/r 


tA 


VECTOR  CALCULUS 


WITH  APPLICATIONS  TO  PHYSICS 


BY 


JAMES  BYRNIE  SHAW 


PROFESSOR  OF  MATHEMATICS  IN  THE  UNIVERSITY  OF  ILLINOIS 


ILLUSTRATED 


NEW  YORK 
D.  VAN    NOSTRAND    COMPANY 

Eight  Warren  Street 
1922 


s 


Copyright,  1922 
By  D.  Van  Nostrand  Company 


All  rights  reserved,  including  that  of  translation  into 
foreign  languages,  including  the  Scandinavian 


Printed  in  the  United  States  of  America 


PREFACE. 

This  volume  embodies  the  lectures  given  on  the  subject 
to  graduate  students  over  a  period  of  four  repetitions.  The 
point  of  view  is  the  result  of  many  years  of  consideration 
of  the  whole  field.  The  author  has  examined  the  various 
methods  that  go  under  the  name  of  Vector,  and  finds  that 
for  all  purposes  of  the  physicist  and  for  most  of  those  of  the 
geometer,  the  use  of  quaternions  is  by  far  the  simplest  in 
theory  and  in  practice.  The  various  points  of  view  are 
mentioned  in  the  introduction,  and  it  is  hoped  that  the  es- 
sential differences  are  brought  out.  The  tables  of  com- 
parative notation  scattered  through  the  text  will  assist  in 
following  the  other  methods. 

The  place  of  vector  work  according  to  the  author  is  in 
the  general  field  of  associative  algebra,  and  every  method  so 
far  proposed  can  be  easily  shown  to  be  an  imperfect  form 
of  associative  algebra.  From  this  standpoint  the  various 
discussions  as  to  the  fundamental  principles  may  be  under- 
stood. As  far  as  the  mere  notations  go,  there  is  not  much 
difference  save  in  the  actual  characters  employed.  These 
have  assumed  a  somewhat  national  character.  It  is  un- 
fortunate that  so  many  exist. 

The  attempt  in  this  book  has  been  to  give  a  text  to  the 
mathematical  student  on  the  one  hand,  in  which  every 
physical  term  beyond  mere  elementary  teims  is  carefully 
defined.  On  the  other  hand  for  the  physical  student  there 
will  be  found  a  large  collection  of  examples  and  exercises 
which  will  show  him  the  utility  of  the  mathematical  meth- 
ods. So  very  little  exists  in  the  numerous  treatments  of 
the  day  that  does  this,  and  so  much  that  is  labeled  vector 

iii 

505384 


IV  PREFACE 

analysis  is  merely  a  kind  of  short-hand,  that  it  has  seemed 
very  desirable  to  show  clearly  the  actual  use  of  vectors  as 
vectors.  It  will  be  rarely  the  case  in  the  text  that  any  use 
of  the  components  of  vectors  will  be  found.  The  triplexes 
in  other  texts  are  very  seldom  much  different  fiom  the  ordi- 
nary Cartesian  forms,  and  not  worth  learning  as  methods. 

The  difficulty  the  author  has  found  with  other  texts  is 
that  after  a  few  very  elementary  notions,  the  mathematical 
student  (and  we  may  add  the  physical  student)  is  suddenly 
plunged  into  the  profundities  of  mathematical  physics,  as 
if  he  were  familiar  with  them.  This  is  rarely  the  case,  and 
the  object  of  this  text  is  to  make  him  familiar  with  them 
by  easy  gradations. 

It  is  not  to  be  expected  that  the  book  will  be  free  from 
errors,  and  the  author  will  esteem  it  a  favor  to  have  all 
errors  and  oversights  brought  to  his  attention.  He  desires 
to  thank  specially  Dr.  C.  F.  Green,  of  the  University  of 
Illinois,  for  his  careful  assistance  in  reading  the  proof,  and 
for  other  useful  suggestions.  Finally  he  has  gathered  his 
material  widely,  and  is  in  debt  to  many  authors  for  it,  to  all 
of  whom  he  presents  his  thanks. 

James  Byrnie  Shaw. 
Urbana,  III., 
July,  1922. 


TABLE  OF  CONTENTS. 

Chapter      I.     Introduction 1 

Chapter     II.     Scalar  Fields 18 

Chapter   III.     Vector  Fields 23 

Chapter    IV.     Addition  of  Vectors 52 

Chapter     V.     Vectors  in  a  Plane 62 

Chapter    VI.     Vectors  in  Space 94 

Chapter  VII.     Applications 127 

1.  The  Scalar  of  two  Vectois 127 

2.  The  Vector  of  two  Vectors 136 

3.  The  Scalar  of  three  Vectors 142 

4.  The  Vector  of  three  Vectors 143 

Chapter  VIII.     Differentials  and  Integrals 145 

1.  Differentiation  as  to  one  Scalar  Parameter ....  145 

Two  Parameters 151 

2.  Differentiation  as  to  a  Vector 155 

3.  Integration 196 

Chapter  IX.     The  Linear  Vector  Function 218 

Chapter   X.     Deformable  Bodies 253 

Strain 253 

Kinematics  of  Displacement 265 

Stress 269 

Chapter  XL     Hydrodynamics 287 


VECTOR  CALCULUS 


CHAPTER  I 
INTRODUCTION 
1.  Vector  Calculus.  By  this  term  is  meant  a  system  of 
mathematical  thinking  which  makes  use  of  a  special  class 
of  symbols  and  their  combinations  according  to  certain 
given  laws,  to  study  the  mathematical  conclusions  resulting 
from  data  which  depend  upon  geometric  entities  called 
vectors,  or  physical  entities  representable  by  vectors,  or 
more  generally  entities  of  any  kind  which  could  be  repre- 
sented for  the  purposes  under  discussion  by  vectors.  These 
vectors  may  be  in  space  of  two  or  three  or  even  four  or 
more  dimensions.  A  geometric  vector  is  a  directed  segment 
of  a  straight  line.  It  has  length  (including  zero)  and  direc- 
tion. This  is  equivalent  to  saying  that  it  cannot  be  de- 
fined merely  by  one  single  numerical  value.  Any  problem 
of  mathematics  dependent  upon  several  variables  becomes 
properly  a  problem  in  vector  calculus.  For  instance, 
analytical  geometry  is  a  crude  kind  of  vector  calculus. 

Several  systems  of  vector  calculus  have  been  devised,  differing  in 
their  fundamental  notions,  their  notation,  and  their  laws  of  combining 
the  symbols.  The  lack  of  a  uniform. notation  is  deplorable,  but  there 
seems  little  hope  of  the  adoption  of  any  uniform  system  soon.  Existing 
systems  have  been  rather  ardently  promoted  by  mathematicians  of  the 
same  nationality  as  their  authors,  and  disagreement  exists  as  to  their 
relative  simplicity,  their  relative  directness,  and  their  relative  logical 
exactness.  These  disagreements  arise  sometimes  merely  with  regard 
to  the  proper  manner  of  representing  certain  combinations  of  the 
symbols,  or  other  matters  which  are  purely  matters  of  convention; 

1 


2  YKCTOR   CALCULUS 

sometimes  they  are  due  to  different  views  as  to  what  are  the  import  an1 
things  to  find  expressions  for;  and  sometimes  they  are  due  to  more 
fundamental  divergences  of  opinion  as  to  the  real  character  of  the 
mathematical  ideas  underlying  any  system  of  this  sort.  We  will  in- 
dicate these  differences  and  dispose  of  them  in  this  work. 

2.  Bases.  We  may  classify  broadly  the  various  systems 
of  vector  calculus  as  geometric  and  algebraic.  The  former 
is  to  be  found  wherever  the  desire  is  to  lay  emphasis  on  the 
spatial  character  of  the  entities  we  are  discussing,  such  as 
the  line,  the  point,  portions  of  a  plane,  etc.  The  latter 
lays  emphasis  on  the  purely  algebraic  character  of  the 
entities  with  which  the  calculations  are  made,  these  entities 
being  similar  to  the  positive  and  negative,  and  the  imag- 
inary of  ordinary  algebra.  For  the  geometric  vector 
systems,  the  symbolism  of  the  calculus  is  really  nothing 
more  than  a  short-hand  to  enable  one  to  follow  certain 
operations  upon  real  geometric  elements,  with  the  possi- 
bility kept  always  in  mind  that  these  entities  and  the 
operations  may  at  any  moment  be  called  to  the  front  to 
take  the  place  of  their  short-hand  representatives.  For 
the  algebraic  systems,  the  symbolism  has  to  do  with 
hypernumbers,  that  is,  extensions  of  the  algebraic  negative 
and  imaginary  numbers,  and  does  not  pretend  to  be  the 
translation  of  actual  operations  which  can  be  made  visible, 
any  more  than  an  ordinary  calculation  of  algebra  could  be 
paralleled  by  actual  geometric  or  physical  operations. 

If  these  distinctions  are  kept  in  mind  the  different  points 
of  view  become  intelligible.  The  best  examples  of  geo- 
metric systems  are  the  Science  of  Extension  of  Grassmann, 
with  its  various  later  forms,  the  Geometry  of  Bynames  of 
Study,  the  Geometry  of  Lines  of  Saussure,  and  the  Geometry 
of  Feuillets  of  Cailler.  The  best  examples  of  algebraic 
systems  are  the  Quaternions  of  Hamilton,  Dyadics  of  Gibbs, 


INTRODUCTION  ,3 

Multenions  of  McAulay,   Biquaternions  of   Clifford,    Tri- 

quaternions  of  Combebiac,   Linear  Associative  Algebra  of 

Peirce.     Various  modifications  of  these  exist,   and  some 

mixed  systems  may  be  found,  which  will  be  noted  in  the 

proper  places. 

The  idea  of  using  a  calculus  of  symbols  for  writing  out  geometric 
theorems  perhaps  originated  with  Leibniz,1  though  what  he  had  in 
mind  had  nothing  to  do  with  vector  calculus  in  its  modern  sense.  The 
first  effective  algebraic  vector  calculus  was  the  Quaternions  of  Hamilton2 
(1843),  the  first  effective  geometric  vector  calculus  was  the  Ausdehn- 
ungslehre  of  Grassmann3  (1844).  They  had  predecessors  worthy  of 
mention  and  some  of  these  will  be  noticed. 

3.  Hypernumbers.  The  real  beginning  of  Vector  Cal- 
culus was  the  early  attempt  to  extend  the  idea  of  number. 
The  original  theory  of  irrational  number  was  metric,4  and 
defined  irrationals  by  means  of  the  segments  of  straight 
lines.  When  to  this  was  added  the  idea  of  direction,  so 
that  the  segments  became  directed  segments,  what  we  now 
call  vectors,  the  numbers  defined  were  not  only  capable  of 
being  irrational,  but  they  also  possessed  quality,  and  could 
be  negative  or  positive.  Ordinary  algebra  is  thus  the  first 
vector  calculus.  If  we  consider  segments  with  direction 
in  a  plane  or  in  space  of  three  dimensions,  then  we  may  call 
the  numbers  they  define  hypernumbers.  The  source  of  the 
idea  was  the  attempt  to  interpret  the  imaginary  which 
had  been  created  to  furnish  solutions  for  any  quadratic  or 
cubic.  The  imaginary  appears  early  in  Cardan's  work.5 
For  instance  he  gives  as  solution  of  the  problem  of  separating 
10  into  two  parts  whose  product  is  40,  the  values 
5  +  V  —  15,  and  5  —  V  —  15.  He  considered  these 
numbers  as  impossible  and  of  no  use.  Later  it  was  dis- 
covered that  in  the  solution  of  the  cubic  by  Cardan's 
formula  there  appeared  the  sum  of  two  of  these  impossible 


4  VWCTOfl  CALCULUS 

values  when  the  answer  actually  was  real.  Bombelli0  #;ive 
as  the  solution  of  the  cubic  r3  =  15x  +  4  the  form 

^(2  +  V  -  121)  +  ^(2  -  V  -  121)  =  4. 

These  impossible  numbers  incited  much  thought  and 
there  came  about  several  attempts  to  account  for  them  and 
to  interpret  them.  The  underlying  question  was  essen- 
tially that  of  existence,  which  at  that  time  was  usually 
sought  for  in  concrete  cases.  The  real  objection  to  the 
negative  number  was  its  inapplicability  to  objects.  Its 
use  in  a  debit  and  credit  account  would  in  this  sense  give  it 
existence.  Likewise  the  imaginary  and  the  complex  num- 
ber, and  later  others,  needed  interpretation,  that  is,  applica- 
tion to  physical  entities. 

4.  Wessel,  a  Danish  surveyor,  in  1797,  produced  a 
satisfactory  method7  of  defining  complex  numbers  by  means 
of  vectors  in  a  plane.  This  same  method  was  later  given 
by  Argand8  and  afterwards  by  Gauss9  in  connection  with 
various  applications.  Wessel  undertook  to  go  farther  and 
in  an  analogous  manner  define  hypernumbers  by  means 
of  directed  segments,  or  vectors,  in  space  of  three  dimen- 
sions. He  narrowly  missed  the  invention  of  quaternions. 
In  1813  Servois10  raised  the  question  whether  such  vectors 
might  not  define  hypernumbers  of  the  form 

.   p  cos  a  +  q  cos  (3  +  r  cos  y 

and  inquired  what  kind  of  non-reals  p,  q,  r  would  be.  He 
did  not  answer  the  question,  however,  and  Wessel's  paper 
remained  unnoticed  for  a  century. 

5.  Hamilton  gave  the  answer  to  the  question  of  Servois 
as  the  result  of  a  long  investigation  of  the  whole  problem.11 
He  first  considered  algebraic  couples,  that  is  to  say  in  our 
terminology,  hypernumbers  needing  two  ordinary  numerical 


INTRODUCTION  5 

values  to  define  them,  and  all  possible  modes  of  combining 
them  under  certain  conditions,  so  as  to  arrive  at  a  similar 
couple  or  hypernumber  for  the  product.  He  then  con- 
sidered triples  and  sets  of  numbers  in  general.  Since  —  1 
and  i  =  V  —  1  are  roots  of  unity,  he  paid  most  attention 
to  definitions  that  would  lead  to  new  roots  of  unity. 

His  fundamental  idea  is  that  the  couple  of  numbers  (a,  b) 
where  a  and  b  are  any  positive  or  negative  numbers,  rational 
or  irrational,  is  an  entity  in  itself  and  is  therefore  subject 
to  laws  of  combination  just  as  are  single  numbers.  For 
instance,  we  may  combine  it  with  the  other  couple  (x,  y) 
in  two  different  ways : 

(a,  b)  +  (x,  y)  =  (a  +  x,  b  +  y) 

(a,  6)  X  (x,  y)  =  {ax  —  by,  ay  +  bx). 

In  the  first  case  we  say  we  have,  added  the  couples,  in  the 
second  case  that  we  have  multiplied  them.  It  is  possible 
to  define  division  also.  In  both  cases  if  we  set  the  couple 
on  the  right  hand  side  equal  to  {u,  v)  we  find  that 

dujdx  —  dv/dy,         dujdy  =  —  dv/dx. 

Pairs  of  functions  u,  v  which  satisfy  these  partial  differential 
equations  Hamilton  called  conjugate  functions.  The  partial 
differential  equations  were  first  given  by  Cauchy  in  this 
connection.     The  particular  couples 

€l   =    (1,   0),  €2   =    (0,    1) 

play  a  special  role  in  the  development,  for,  in  the  first 
place,  any  couple  may  be  written  in  the  form 

(a,  b)  =  aei  +  be2 

and  the  notation  of  couples  becomes  superfluous;  in  the 
second  place,  by  defining  the  products  of  ei  and  e2  in  various 
ways  we  arrive  at  various  algebras  of  couples.     The  general 


C>  VECTOR   CALCULUS 

definition  would  be,  using  the  •  for  X, 

€l'€i   =    Cin€i  +   Cii2€2,  €i'€2  =    Ci2i€i  +    ^12262, 

€2'€i  =    C2ll€i  +  C212€2>  «2  *  €2   =    C221«l  +  C222€2- 

By  varying  the  choice  of  the  arbitrary  constants  c,  and 
Hamilton  considered  several  different  cases,  different 
algebras  of  couples  could  be  produced.  In  the  case  above 
the  c's  are  all  zero  except 

Cm  =  1,         C122  =  1,         C212  —  1,         C221  =  —  1. 

From  the  character  of  4  it  may  be  regarded  as  entirely 
identical  with  ordinary  1,  and  it  follows  therefore  that 
e2  may  be  regarded  as  identical  with  the  V  —  1.  On  the 
other  hand  we  may  consider  €1  to  be  a  unit  vector  pointing 
to  the  right  in  the  plane  of  vectors,  and  c2  to  be  a  unit 
vector  perpendicular  to  ei.  We  have  then  a  vector  calculus 
practically  identical  with  Wessel's.  The  great  merit  of 
Hamilton's  investigation  lies  of  course  in  its  generality. 
He  continued  the  study  of  couples  by  a  similar  study  of 
triples  and  then  quadruples,  arriving  thus  at  Quaternions. 
His  chief  difference  in  point  of  view  from  those  who  followed 
him  and  who  used  the  concept  of  couple,  triple,  etc.  {Mul- 
tiple we  will  say  for  the  general  case),  is  that  he  invariably 
defined  one  product,  whereas  others  define  usually  several. 
6.  Multiples.  There  is  a  considerable  tendency  in  the 
current  literature  of  vector  calculus  to  use  the  notion  of 
multiple.  A  vector  is  usually  designated  by  a  triple  as 
(x,  y,  z),  and  usually  such  triple  is  called  a  vector.  It  is 
generally  tacitly  understood  that  the  dimensions  of  the 
numbers  of  the  triple  are  the  same,  and  in  fact  most  of  the 
products  defined  would  have  no  meaning  unless  this 
homogeneity  of  dimension  were  assumed  to  hold.  We 
find  products  defined  arbitrarily  in  several  ways.  For 
instance,  the  scalar  product  of  the  triples  (a,  b,  c)  and  (x,  y,  z) 


INTRODUCTION  7 

is  =fc  (ax  +  by  +  cz),  the  sign  depending  upon  the  person 
giving  the  definition;  the  vector  product  of  the  same  two 
triples  is  usually  given  as  the  triple  (bz  —  cy,  ex  —  az, 
ay  —  bx).  It  is  obvious  at  once  that  a  great  defect  of  such 
definitions  is  that  the  triples  involved  have  no  sense  until 
the  significance  of  the  first  number,  the  second  number, 
and  the  third  number  in  each  triple  is  understood.  If 
these  depend  upon  axes  for  their  meaning,  then  the  whole 
calculus  is  tied  down  to  such  axes,  unless,  as  is  usually 
done,  the  expressions  used  in  the  definitions  are  so  chosen 
as  to  be  in  some  respects  independent  of  the  particular 
set  of  axes  chosen.  When  these  expressions  are  thus 
chosen  as  invariants  under  given  transformations  of  the 
axes  we  arrive  at  certain  of  the  well-known  systems  of 
vector  analysis.  The  transformations  usually  selected  to 
furnish  the  profitable  expressions  are  the  group  of  orthog- 
onal transformations.  For  instance,  it  was  shown  by 
Burkhardt12  that  all  the  invariant  expressions  or  invariant 
triples  are  combinations  of  the  three  following : 

ax  +  by  +  cz, 

(bz  —  cy,  ex  —  az,  ay  —  bx), 

(al  +  bm  +  cn)x  +  (am  —  bl)y  +  (an  —  cl)z, 

(bl  —  am)x  +  (al  -f-  bm  +  cn)y  +  (bn  —  cm)z, 

(cl  —  ari)x  +  (cm  —  bn)u  +  (al  +  bm  +  cn)z. 

A  study  of  vector  systems  from  this  point  of  view  has 
been  made  by  Schouten.13 

7.  Quaternions.  In  his  first  investigations,  Hamilton 
was  chiefly  concerned  with  the  creation  of  systems  of 
hypernumbers  such  that  each  of  the  defining  units,  similar 
to  the  ei  and  €2  above,  was  a  root  of  unity.14  That  is,  the 
process  of  multiplication  by  iteration  would  bring  back  the 
multiplicand.    He  was  actually  interested  in  certain  special 


8  VECTOR  CALCULUS 

cases  of  abstract  groups,15  and  if  he  had  noticed  the  group 
property  his  researches  would  perhaps  have  extended  into 
the  whole  field  of  abstract  groups.  In  quaternions  he  found 
a  set  of  square  roots  of  —  1,  which  he  designated  by  i,  j,  k, 
connected  with  his  triples  though  belonging  to  a  set  of  quad- 
ruples. In  his  Lectures  on  Quaternions,  the  first  treatise  he 
published  on  the  subject,  he  chose  a  geometrical  method  of 
exposition,  consequently  many  have  been  led  to  think  of 
quaternions  as  having  a  geometric  origin.  However,  the 
original  memoirs  show  that  they  were  reached  in  a  purely 
algebraic  way,  and  indeed  according  to  Hamilton's  philoso- 
phy were  based  on  steps  of  time  as  opposed  to  geometric 
steps  or  vectors. 

The  geometric  definition  is  quite  simple,  however,  and 
not  so  abstract  as  the  purely  algebraic  definition.  Ac- 
cording to  this  idea,  numbers  have  a  metric  definition,  a 
number,  or  hypernumber,  being  the  ratio  of  two  vectors. 
If  the  vectors  have  the  same  direction  we  arrive  at  the 
ordinary  numerical  scale.  If  they  are  opposite  we  arrive 
at  the  negative  numbers.  If  neither  in  the  same  direction 
nor  opposite  we  have  a  more  general  kind  of  number,  a 
hypernumber  in  fact,  which  is  a  quaternion,  and  of  which 
the  ordinary  numbers  and  the  negative  numbers  are 
merely  special  cases.  If  we  agree  to  consider  all  vectors 
which  are  parallel  and  in  the  same  direction  as  equivalent, 
that  is,  call  them  free  vectors,  then  for  every  pair  of  vectors 
from  the  origin  or  any  fixed  point,  there  is  a  quaternion. 
Among  these  quaternions  relations  will  exist,  which  will 
be  one  of  the  objects  of  study  of  later  chapters. 

8.  Mobius  was  one  of  the  early  inventors  of  a  vector 
calculus  on  the  geometric  basis.  In  his  Barycentrisch.es 
Kalkul16  he  introduced  a  method  of  deriving  points  from 
other   points   by   a   process   called   addition,   and   several 


INTRODUCTION  9 

applications  were  made  to  geometry.  The  barycentric 
calculus  is  somewhat  between  a  system  of  homogeneous 
coordinates  and  a  real  vector  calculus.  His  addition  was 
used  by  Grassmann. 

9.  Grassmann  in  1844  published  his  treatise  called  Die 
lineale  Ausdehnungslehre17  in  which  several  different  proc- 
esses called  multiplication  are  used  for  the  derivation  of 
geometric  entities  from  other  geometric  entities.  These 
processes  make  use  of  a  notation  which  is  practically  a 
sort  of  short-hand  for  the  geometric  processes  involved. 
Grassmann  considered  these  various  kinds  of  multiplication 
abstractly,  leaving  out  of  account  the  meaning  of  the 
elements  multiplied.  His  methods  apply  to  space  of  N 
dimensions.  In  the  symmetric  multiplication  it  is  possible 
to  interchange  any  two  of  the  factors  without  affecting  the 
result.  In  the  circular  multiplication  the  order  may  be 
changed  cyclically.  In  the  lineal  multiplication  all  the 
laws  hold  as  well  for  any  factors  which  are  linear  combina- 
tions of  the  hypernumbers  which  define  the  base,  as  for 
those  called  the  base.  He  studies  two  species  of  circular 
multiplication.  If  the  defining  units  of  the  base  are  ex,  e2,  e3 
•  •  •€„,  then  we  have  in  the  first  variety  of  circular  multipli- 
cation the  laws 

€l2  +  €22  +  632  +    •  •  •   +  €n2  =    0,  €i€j  =   €j€i. 

In  the  second  variety  we  have  the  laws 

ei2  =  0,        e/  =  0,  •  -  •  en2  =  0,        Mi  =  0,        *+j. 

In  the  lineal  genus  of  multiplication  he  studies  two 
species,  in  the  first,  called  the  algebraic  multiplication,  we 
have  the  law 

My  =  *fii         for  all  i,  j. 

while  in  the  second,  called  the  exterior  multiplication,  the 
interchange  of  any  two  factors  changes  the  sign  of  the 


10  VECTOR  CALCULUS 

result.  Of  the  latter  there  are  two  varieties,  the  progressive 
multiplication  in  which  the  number  of  dimensions  of  the 
geometric  figure  which  is  the  product  is  the  sum  of  the 
dimensions  of  the  factors,  while  in  the  other,  called  re- 
gressive multiplication,  the  dimension  of  the  product  is  the 
difference  between  the  sum  of  the  dimensions  of  the  factors 
and  N  the  dimension  of  the  space  in  which  the  operation 
takes  place.  From  the  two  varieties  he  deduces  another 
kind  called  interior  multiplication. 

If  we  confine  our  thoughts  to  space  of  three  dimensions, 
defined  by  points,  and  if  €1,  e2,  e3,  e4  are  such  points,  the 
progressive  exterior  product  of  two,  as  €1,  e2,  is  ei€2  and 
represents  the  segment  joining  them  if  they  do  not  coincide. 
The  product  is  zero  if  they  coincide.  The  product  of  this 
into  a  third  point  €3  is  ei€2e3  and  represents  the  parallelogram 
with  edges  €162,  ei€3  and  the  other  two  parallel  to  these 
respectively.  If  all  three  points  are  in  a  straight  line  the 
product  is  zero.  The  exterior  progressive  product  c1e2e3€4 
represents  the  parallelepiped  with  edges  €ie2,  €ie3,  €i€4  and 
the  opposite  parallel  edges.  The  regressive  exterior  product 
of  €i€2  and  €ie3€4  is  their  common  point  €1.  The  regressive 
product  of  €ie2e3  and  €ie2€4  is  their  common  line  €ie2.  The 
complement  of  €1  is  defined  to  be  €2e3e4,  and  of  €i€2  is  e3fct, 
and  of  €i€2e3  is  €4.  The  interior  product  of  any  expression 
and  another  is  the  progressive  or  regressive  product  of  the 
first  into  the  complement  of  the  other.  For  instance,  the 
interior  product  of  €1  and  e2  is  the  progressive  product  of 
€1  and  €i€3e4  which  vanishes.  The  interior  product  of  e2 
and  e2  is  the  product  of  e2  and  eie3e4  which  is  €2eie3e4.  The 
interior  product  of  €j€2e3  and  ei€4  is  the  product  of  €ie2e3 
and  €2e3  which  would  be  regressive  and  be  the  line  e2e3. 

We  have  the  same  kinds  of  multiplication  if  the  expres- 
sions e  are  vectors  and  not  points,  and  they  may  even  be 


INTRODUCTION  1 1 

planes.  The  interpretation  is  different,  however.  It  is 
easy  to  see  that  Grassmann's  ideas  do  not  lend  themselves 
readily  to  numerical  application,  as  they  are  more  closely 
related  to  the  projective  transformations  of  space.  In 
fact,  when  translated,  most  of  the  expressions  would  be 
phrased  in  terms  of  intersections,  points,  lines  and  planes, 
rather  than  in  terms  of  distances,  angles,  areas,  etc. 

10.  Dyadics  were  invented  by  Gibbs,18  and  are  of  both  the 
algebraic  and  the  geometric  character.  Gibbs  has,  like 
Hamilton,  but  one  kind  of  multiplication.  If  we  have 
given  two  vectors  a,  (3  from  the  same  point,  their  dyad  is  a(3. 
This  is  to  be  looked  upon  as  a  new  entity  of  two  dimensions 
belonging  to  the  point  from  which  the  vectors  are  drawn. 
It  is  not  a  plane  though  it  has  two  dimensions,  but  is  really 
a  particular  and  special  kind  of  dyadic,  an  entity  of  two- 
dimensional  character,  such  that  in  every  case  it  can  be 
considered  to  be  the  sum  of  not  more  than  three  dyads. 
Gibbs  never  laid  any  stress  on  the  geometric  existence  of 
the  dyadic,  though  he  stated  definitely  that  it  was  to  be 
considered  as  a  quantity.  His  greatest  stress,  however, 
was  upon  the  operative  character  of  the  dyadic,  its  various 
combinations  with  vectors  being  easily  interpretable.  The 
simplest  interpretation  is  from  its  use  in  physics  to  represent 
strain. 

Gibbs  also  pushed  his  vector  calculus  into  space  of  many 
dimensions,  and  into  triadic  and  higher  forms,  most  of 
which  can  be  used  in  the  theory  of  the  elasticity  of  crystals. 
The  scalar  and  vector  multiplication  he  considered  as 
functions  of  the  dyadic,  rather  than  as  multiplications, 
and  there  are  corresponding  functions  of  triadics  and 
higher  forms.  In  this  respect  his  point  of  view  is  close  to 
that  of  Hamilton,  the  difference  being  in  the  use  of  the 
dyadic  or  the  quaternion. 

11.  Other  forms  of  vector  calculus  can  be  reduced  to 
3 


12  VECTOR  CALCULUS 

these  or  to  combinations  of  parts  of  these.     The  differences 
are  usually  in  the  notations,  or  in  the  basis  of  exposition. 

Notations  for  One  Vector 

Greek  letters,  Hamilton,  Tait,  Joly,  Gibbs. 

Italics,  Grassmann,_Peano,  Fehr,  Ferraris,  Macfarlane. 
Heun  writes  a,  b,  c. 

Old  English  or  German  letters,  Maxwell,  Jaumann,  Jung, 
Foppl,  Lorentz,  Gans,  Abraham,  Bucherer,  Fischer, 
Sommerfeld. 

Clarendon  type,  Heaviside,  Gibbs,  Wilson,  Jahnke,  Timer- 
ding,  Burali-Forti,  Marcolongo. 

Length  of  a  vector 
T  ( ),  Hamilton,  Tait,  Joly. 
|  | ,  Gans,  Bucherer,  Timerding. 
Italic  corresponding  to  the  ve  ctor  letter,  Wilson,  Jaumann, 

&ing,    Fischer,    Jahnke.     Corresponding    small    italic, 

Macfarlane. 
Mod.  (  ),  Peano,  Burali-Forti,  Marcolongo,  Fehr. 

Unit  of  a  vector 
U  ( ),  Hamilton,  Tait,  Joly,  Peano. 
Clarendon  small,  Wilson. 
(  )i,  Bucherer,  Fischer. 
Corresponding  Greek  letter,  Macfarlane. 
Some  write  the  vector  over  the  length. 

Square  of  a  vector 
(  )2.     The  square  is  usually  positive  except  in  Quaternions, 
where  it  is  negative. 

Reciprocal 

( )-1,  Hamilton,  Tait,  Joly,  Jaumann. 

tt  ,  Hamilton,  Tait,  Joly,  Fischer,  Bucherer. 


CHAPTER  II 

SCALAR  FIELDS 
1.  Fields.  If  we  consider  a  given  set  of  elements  in 
space,  we  may  have  for  each  element  one  or  more  quantities 
determined,  which  can  be  properly  called  functions  of  the 
element.  For  instance,  at  each  point  in  space  we  may  have 
a  temperature,  or  a  pressure,  or  a  density,  as  of  the  air. 
Or  for  every  loop  that  we  may  draw  in  a  given  space  we 
may  have  a  length,  or  at  some  fixed  point  a  potential  due 
to  the  loop.  Again,  we  may  have  at  each  point  in  space 
a  velocity  which  has  both  direction  and  length,  or  an 
electric  intensity,  or  a  magnetic  intensity.  Not  to  multiply 
examples  unnecessarily,  we  can  see  that  for  a  given  range 
of  points,  or  lines,  or  other  geometric  elements,  we  may 
have  a  set  of  quantities,  corresponding  to  the  various 
elements  of  the  range,  and  therefore  constituting  a  function 
of  the  range,  and  these  quantities  may  consist  of  numerical 
values,  or  of  vectors,  or  of  other  hypernumbers.  When 
they  are  of  a  simple  numerical  character  they  are  called 
scalars,  and  the  function  resulting  is  a  scalar  function. 
Examples  are  the  density  of  a  fluid  at  each  point,  the  density 
of  a  distribution  of  energy,  and  similar  quantities  consisting 
of  an  amount  of  some  entity  per  cubic  centimeter,  or  per 
square  centimeter,  or  per  centimeter. 

EXAMPLES 
(1)  Electricity.     The  unit  of   electricity  is   the  coulomb, 
connected  with  the  absolute  units  by  the  equations 
1  coulomb  =  3  •  10°  electrostatic  units 

==  10-1  electromagnetic  units. 
13 


14  VECTOR  CALCULUS 

The  density  of  electricity  is  its  amount  in  a  given  volume, 
area,  op  length  divided  by  the  volume,  area,  or  length 
respectively.  The  dimensions  of  electricity  will  be  repre- 
sented by  [9],  and  for  its  amount  the  symbol  9  will  be  used. 
For  the  volume  density  we  will  use  e,  for  areal  density  e' , 
for  linear  density  e".  If  the  distribution  may  be  considered 
to  be  continuous,  we  may  take  the  limits  and  find  the 
density  at  a  point. 

(2)  Magnetism.  Considering  magnetism  to  be  a  quan- 
tity, we  will  use  for  the  unit  of  measurement  the  maxwell, 
connected  with  the  absolute  units  by  the  equation 

1  maxwell  =  3-1010  electrostatic  units 

=  1  electromagnetic  unit. 

Sometimes  108  maxwells  is  called  a  weber.  The  symbol  for 
magnetism  will  be  $,  the  dimensions  [$],  the  densities 
m,  m',  m". 

(3)  Action.  This  quantity  is  much  used  in  physics,  the 
principle  of  least  action  being  one  of  the  most  important 
fundamental  bases  of  modern  physics.  The  dimensions 
of  action  are  [93>],  the  symbol  we  shall  use  is  A,  and  the 
unit  might  be  a  quantum,  but  for  practical  purposes  a 
joule-second  is  used.  In  the  case  of  a  moving  particle  the 
action  at  any  point  depends  upon  the  path  by  which  the 
particle  has  reached  the  point,  so  that  as  a  function  of  the 
points  of  space  it  has  at  each  point  an  infinity  of  values. 
A  function  which  has  but  a  single  value  at  a  point  will  be 
called  monodromic,  but  if  it  has  more  than  one  value  it  will 
be  called  polydromic.  The  action  is  therefore  a  polydromic 
function.  We  not  only  have  action  in  the  motion  of  par- 
ticles but  we  find  it  as  a  necessary  function  of  a  momentum 
field,  or  of  an  electromagnetic  field. 

(4)  Energy.     The  unit  of  energy  is  the  erg  or  the  joule 


SCALAR   FIELDS  15 

=  107  ergs.     Its  dimensions  are    [G^T7-1],  its  symbol  will 
beW. 

(5)  Activity.  This  should  not  be  confused  with  action. 
It  is  measured  in  watts,  symbol  J,  dimensions  [Q$T~2]. 

(6)  Energy-density.     The  symbol  will  be  U,  dimensions 

(7)  Activity-density.     The  symbol  will  be  Q,  dimensions 

pi-3r2]. 

(8)  Mass.  The  symbol  is  M,  dimensions  [0$77r2]. 
The  unit  of  mass  is  the  gram.  A  distribution  of  mass  is 
usually  called  a  distribution  of  matter. 

(9)  Density  of  mass.     The  symbol  will  be  c,  dimensions 

(10)  Potential    of    electricity.     Symbol     V,    dimensions 

(11)  Potential  of  magnetism.  Symbol  N,  dimensions 
[027-1]. 

(12)  Potential  of  gravity.    Symbol  P,  dimensions  [G^T7-1]. 

2.  Levels.  Points  at  which  the  function  has  the  same 
value,  are  said  to  define  a  level  surface  of  the  function.  It 
may  have  one  or  more  sheets.  Such  surfaces  are  usually 
named  by  the  use  of  the  prefixes  iso  and  equi.  For  instance, 
the  surfaces  in  a  cloud,  which  have  all  points  at  the  same 
temperature,  are  called  isothermal  surfaces;  surfaces  which 
have  points  at  the  same  pressure  are  called  isobaric  surfaces; 
surfaces  of  equal  density  are  isopycnic  surfaces;  those  of 
equal  specific  volume  (reciprocal  of  the  density)  are  the  iso- 
steric  surfaces;  those  of  equal  humidity  are  isohydric  surfaces. 
Likewise  for  gravity,  electricity,  and  magnetism  we  have 
equipotential  surfaces. 

3.  Lamellae.  Surfaces  are  frequently  considered  for 
which  we  have  unit  difference  between  the  values  of  the 
function  for  the  successive  surfaces.     These  surfaces  and 


16  VECTOR   CALCULUS 

the  space  between  them  constitute  a  succession  of  unit 
lamellae. 

If  we  follow  a  line  from  a  point  A  to  a  point  B,  the  number 
of  unit  lamellae  traversed  will  give  the  difference  between 
the  two  values  of  the  function  at  the  points  A  and  B. 
If  this  is  divided  by  the  length  of  the  path  we  shall  have  the 
mean  rate  of  change  of  the  function  along  the  path.  If 
the  path  is  straight  and  the  unit  determining  the  lamellae 
is  made  to  decrease  indefinitely,  the  limit  of  this  quotient 
at  any  point  is  called  the  derivative  of  the  function  at 
that  point  in  the  given  direction.  The  derivative  is  ap- 
proximately the  number  of  unit  lamellae  traversed  in  a 
unit  distance,  if  they  are  close  together. 

4.  Geometric  Properties.  Monodromic  levels  cannot  in- 
tersect each  other,  though  any  one  may  intersect  itself. 
Any  one  or  all  of  the  levels  may  have  nodal  lines,  conical 
points,  pinch-points,  and  the  other  peculiarities  of  geo- 
metric surfaces.  These  singularities  usually  depend  upon 
the  singularities  of  the  congruence  of  normals  to  the 
surface. 

In  the  case  of  functions  of  two  variables,  the  scalar  levels 
will  be  curves  on  the  surface  over  which  the  two  variables 
are  defined.  Their  singularities  may  be  any  that  can 
occur  in  curves  on  surfaces. 

5.  Gradient.  The  equation  of  a  level  surface  is  found 
by  setting  the  function  equal  to  a  constant.  If,  for  in- 
stance, the  point  is  located  by  the  coordinates  x,  y,  z 
and  the  function  is  f(x,  y,  z),  then  the  equation  of  any 
level  is 

u  =  /(*>  V>  z)  =  C. 

If  we  pass  to  a  neighboring  point  on  the  same  surface 
we  have 

du  =  f{x  +  dx,  y  -f-  dy,  z  +  dz)  —  f{x,  y,  z)  =  0. 

We  may  usually  find  functions  df/dx,  bf\a\  df/dz, 


SCALAR   FIELDS  17 

functions  independent  of  dx,  dy,  dz,  such  that 

du  —  dfjdx  •  dx  +  df/dy  •  dy  +  df/dz  •  dz. 

Now  the  vector  from  the  first  point  to  the  second  has 
as  the  lengths  of  its  projections  on  the  axes:  dx,  dy,  dz;  and 
if  we  define  a  vector  whose  projections  are  dfjdx,  df/dy, 
df/dz,  which  we  will  call  the  Gradient  of  f,  then  the  con- 
dition du  =  0  is  the  condition  that  the  gradient  of  /  shall  be 
perpendicular  to  the  differential  on  the  surface.  Hence, 
if  we  represent  the  gradient  of  /  by  v/,  and  the  differential 
change  from  one  point  to  the  other  by  dp,  we  see  that  dp 
is  any  infinitesimal  tangent  on  the  surface  and  v/  is  along  the 
normal  to  the  surface.  It  is  easy  to  see  that  if  we  differen- 
tiate u  in  a  direction  not  tangent  to  a  level  surface  of  u  we 
shall  have 

du  =  df/dx-dx  +  df/dy  •<&,+  df/dz -dz  =  dC. 

If  the  length  of  the  differential  path  is  ds  then  we  shall  have* 
du/ds  =  projection  of^fon  the  unit  vector  in  the  direction  of  dp. 
The  length  of  the  vector  v/  is  sometimes  called  the  gradient 
rather  than  the  vector  itself.  Sometimes  the  negative  of 
the  expression  used  here  is  called  the  gradient. 

When  the  three  partial  derivatives  of  /  vanish  for  the 
same  point,  the  intensity  of  the  gradient,  measured  by  its 
length,  is  zero,  and  the  direction  becomes  indeterminate 
from  the  first  differentials.  At  such  points  there  are  singu- 
larities of  the  function.  At  points  where  the  function 
becomes  infinite,  the  gradient  becomes  indeterminate  and 
such  points  are  also  singular  points. 

6.  Potentials.  The  three  components  of  a  vector  at  a 
point  may  be  the  three  partial  derivatives  of  the  same 
function  as  to  the  coordinates,  in  which  case  the  vector 
may  be  looked  upon  as  the  gradient  of  the  integral  func- 

*  Since  dxjds,  dyjds,  dzjds  are  the  direction-cosines  of  dp. 


18  VECTOR  CALCULUS 

tion,  which  is  called  a  potential  junction,  or  sometimes  a 
force  function.  For  instance,  if  the  components  of  the 
velocity  satisfy  the  proper  conditions,  the  velocity  is  the 
gradient  of  a  velocity  'potential.  These  conditions  will  be 
discussed  later,  and  the  vector  will  be  freed  from  dependence 
upon  any  axes. 

7.  Relative  Derivatives.  In  case  there  are  two  scalar 
functions  at  a  point,  we  may  have  use  for  the  concept  of 
the  derivative  of  one  with  respect  to  the  other.  This  is 
defined  to  be  the  quotient  of  the  intensity  of  the  gradient  of 
the  first  by  that  of  the  second,  multiplied  by  the  cosine 
of  their  included  angle.  If  the  unit  lamellae  are  constructed, 
it  is  easy  to  see  from  the  definition  that  the  relative  deriva- 
tive of  the  first  as  to  the  second  will  be  the  limit  of  the 
average  or  mean  of  the  number  of  unit  sheets  of  the  first 
traversed  from  one  point  to  another,  along  the  normal  of  the 
second  divided  by  the  number  of  unit  sheets  of  the  second 
traversed  at  the  same  time.  For  instance,  if  we  draw  the 
isobars  for  a  given  region  of  the  United  States  and  the 
simultaneous  isotherms,  then  in  passing  from  a  point  A 
to  a  point  B  if  we  traverse  24  isobaric  unit  sheets  and  10 
isothermal  unit  sheets,  the  average  is  2.4  isobars  per 
isotherm.  ^ 

8.  Unit-Tubes.  If  there  are  two  scalar  functions  in  the 
field,  and  the  unit  lamellae  are  drawn,  the  unit  sheets  will 
usually  intersect  so  as  to  divide  the  space  under  considera- 
tion into  tubes  whose  cross-section  will  be  a  curvilinear 
parallelogram.  Since  the  area  of  such  parallelogram  is 
approximately 

dsids2  esc  0, 
where  dsi  is  the  distance  from  a  unit  sheet  of  the  function  u 
to  the  next  unit  sheet,  and  ds2  the  corresponding  distance 
for  the  function  v,  while  6  is  the  angle  between  the  surfaces; 
and  since  we  have,  Tyu  being  the  intensity  of  the  gradient 


SCALAR   FIELDS  19 

of  u,  and  T^/v  the  intensity  of  the  gradient  of  v, 

dsi  -  1/TVu,         ds2  =  1/Tw 

the  area  of  the  parallelogram  will  be  l/(TyuTvv  sin  6). 
Consequently  if  we  count  the  parallelograms  in  any  plane 


Fig.  1. 

cross-section  of  the  two  sets  of  level  surfaces,  this  number 
is  an  approximate  value  of  the  expression 

T^uT^Jv  sin  6  X  area  parallelogram 

when  summed  over  the  plane  cross-section.  That  is  to 
say,  the  number  of  these  tubes  which  stand  perpendicular 
to  the  plane  cross-section  is  the  approximate  integral  of  the 
expression  T^uT^v  sin  6  over  the  area  of  the  cross-section. 
These  tubes  are  called  unit  tubes  for  the  same  reason  that 
the  lamellae  are  called  unit  lamellae. 

In  counting  the  tubes  it  must  be  noticed  whether  the 
successive  surfaces  crossed  correspond  to  an  increasing  or 
to  a  decreasing  value  of  u  or  of  v.  It  is  also  clear  that 
when  sin  6  is  everywhere  0  the  integral  must  be  zero.  In 
such  case  the  three  Jacobians 

d(u,  v)/d(y,  z),         d(u,  v)/d(z,  x),         d{u,  v)/d(x,  y) 


20  VECTOR  CALCULUS 

are  each  equal  zero,  and  this  is  the^condition  that  u  is  a 
function  of  v.  In  case  the  plane  of  cross-section  is  the 
x,  y  plane,  the  first  two  expressions  vanish  anyhow,  since 
u,  v  are  functions  of  x,  y  only. 

It  is  clear  if  we  take  the  levels  of  one  of  the  functions, 
say  u,  as  the  upper  and  lower  parts  of  the  boundary  of  the 
cross-section,  that  in  passing  from  one  of  the  other  sides 
of  the  boundary  along  each  level  of  u  the  number  of  unit 
tubes  we  encounter  from  that  side  of  the  boundary  to  the 
opposite  side  is  the  excess  of  the  value  of  v  on  the  second 
side  over  that  on  the  first  side.  If  then  we  count  the  dif- 
ferent tubes  in  the  successive  lamellae  of  u  between  the 
two  sides  of  the  cross-section  we  shall  have  the  total  excess 
of  those  on  the  second  side  over  those  on  the  first  side. 
That  is  to  say,  the  number  of  unit  tubes  or  the  integral 
over  the  area  bounded  by  level  1  and  level  2  of  u,  and  any 
other  two  lines  which  cross  these  two  levels  so  as  to  produce 
a  simple  area  between,  is  the  excess  of  the  sum  between 
the  two  levels  of  the  values  of  v  on  one  side  over  the  same 
sum  between  the  two  levels  of  u  on  the  other  side.  These 
graphical  solutions  are  used  in  Meteorology. 

This  gives  the  excess   of  the  integral   J  vdu  along  the 

second  line  between  the  two  levels  of  u,  over  the  same  in- 
tegral along  the  first  line.  It  represents  the  increase  of  this 
integral  in  a  change  of  path  from  one  line  to  the  other.  For 
instance  if  the  integral  is  energy,  the  number  of  tubes  is 
the  amount  of  energy  stored  or  released  in  the  passage  from 
one  line  to  the  other,  as  in  a  cyclone.  The  number  of  tubes 

for   any  closed  path  is   the   approximate    integral    I  rdu 

around  the  path.  , 


SCALAR   FIELDS  21 

EXERCISES. 

1.  If  the  density  varies  as  the  distance  from  a  given  axis,  what  are 
the  isopycnic  surfaces? 

2.  A  rotating  fluid  mass  is  in  equilibrium  under  the  force  of  gravity, 
the  hydrostatic  pressure,  and  the  centrifugal  force.  What  are  the 
levels?     Show  that  the  field  of  force  is  conservative. 

3.  The  isobaric  surfaces  are  parallel  planes,  and  the  isopycnic 
surfaces  are  parallel  planes  at  an  angle  of  10°  with  the  isobaric  planes. 
What  is  the  rate  of  change  of  pressure  per  unit  rate  of  change  of  density 
along  a  line  at  45°  with  the  isobaric  planes? 

4.  If  the  pressure  can  be  stated  as  a  function  of  the  density,  what 
conditions  are  necessary?  Are  they  sufficient?  What  is  the  interpreta- 
tion with  regard  to  the  levels? 

5.  Three  scalar  functions  have  a  functional  relation  if  their  Jacobian 
vanishes.     What  does  this  mean  with  regard  to  their  respective  levels? 

6.  If  the  isothermal  surfaces  are  spheres  with  center  at  the  earth's 
center,  the  temperature  sheets  for  decrease  of  one  degree  being  166.66 
feet  apart,  and  if  the  isobaric  levels  are  similar  spheres,  the  pressure 
being  given  by 

log  B  =  log  B,  -  0.0000177 (a  -  z0), 

where  B0  is  the  pressure  at  z0  feet  above  the  surface  of  the  earth,  what 
is  the  relative  derivative  of  the  temperature  as  to  the  pressure,  and  the 
pressure  as  to  the  temperature? 

7.  To  find  the  maximum  of  u(x,  y,  z)  we  set  du  =  0.  If  there  is  also 
a  condition  to  be  fulfilled,  v(x,  y,  z)  =  0,  then  dv  =  0  also. 

These  two  equations  in  dx,  dy,  dz  must  be  satisfied  for  all  compatible 
values  of  dx,  dy,  dz,  and  we  must  therefore  have 

du    du    du  _  _  dy  #  dv    dv_ 
dx'  dy'   dz'   ~  dx'  dy'  dz} 

which  is  equivalent  to  the  single  vector  equation 

Vw  =  wyv. 

What  does  this  mean  in  terms  of  the  levels :;     The  unit  tubes? 

If  there  is  also  another  equation  of  condition  l(x,  y,  z)  =0  then  also 
dt  =  0  and  the  Jacobian  of  the  three  functions  u,  v,  t  must  equal  zero. 
Interpret. 

8.  On  the  line  of  intersection  of  two  levels  of  two  different  functions 
the  values  of  both  functions  remain  constant.  If  we  differentiate  a 
third  function  along  the  locus  in  question,  the  differential  vanishing 
everywhere,  what  is  the  significance? 


22 


VECTOR   CALCULUS 


9.  If  a  field  of  force  has  a  potential,  then  a  fluid,  subject  to  the  force 
and  such  that  its  pressure  is  a  function  of  the  density  and  the  tempera- 
ture, will  have  the  equipotential  levels  for  isobaric  levels  also.  The 
density  will  be  the  derivative  of  the  pressure  relative  to  the  potential. 
Show  therefore  that  equilibrium  is  not  possible  unless  the  isothermals 
are  also  the  levels  of  force  and  of  pressure. 

[p  =  p(c,  T),  and  vp  =  cvv  =  PcVc  +  prvT. 

If  then  vc  =  0,  cvv  =  prVT.] 

10.  If  the  full  lines  below  represent  the  profiles  of  isobaric  sheets,  and 
the  dotted  lines  the  profiles  of  isosteric  sheets,  count  the  unit  tubes 
between  the  two  verticals,  and  explain  what  the  number  means.  If 
they  were  equipotentials  of  gravity  and  isopycnic  surfaces,  what  would 
the  number  of  unit  tubes  mean? 


Fig.  2. 

11.  If  u  =  y  —  12x3  and  v  =  y  +  x2  +  \x,  find  Vw  and  w  and 
TvuTw -sin  6,  and  integrate  the  latter  over  the  area  between  x  =  0f 
x  =  1,  y  =  0,  y  =  12.     Draw  the  lines. 

12.  If  u  =  ax  +  by  +  cz  and  v  =  x2  -f-  if  +  z2,  find  vw  and  vv  and 
TyuTvvsm  6  and  integrate  the  latter  expression  over  the  surface  of  a 
cylinder  whose  axis  is  in  the  direction  of  the  z  axis.  Find  the  deriva- 
tive of  each  relative  to  the  other. 


CHAPTER  III 

VECTOR  FIELDS 

1.  Hypercomplex  Quantity.  In  the  measurement  of 
quantity  the  first  and  most  natural  invention  of  the  mind 
was  the  ordinary  system  of  integers.  Following  this  came 
the  invention  of  fractions,  then  of  irrational  numbers. 
With  these  the  necessary  list  of  numbers  for  mere  measure- 
ment of  similar  quantities  is  closed,  up  to  the  present  time. 
Whether  it  will  be  necessary  to  invent  a  further  extension 
of  number  along  this  line  remains  for  the  future  to  show. 

In  the  attempt  to  solve  equations  involving  ordinary 
numbers,  it  became  necessary  to  invent  negative  numbers 
and  imaginary  numbers.  These  were  known  and  used  as 
fictitious  numbers  before  it  was  noticed  that  quantities 
also  are  of  a  negative  or  an  "imaginary"  character.  We 
find  instances  everywhere.  In  debit  and  credit,  for  ex- 
ample, we  have  quantity  which  may  be  looked  upon  as  of 
two  different  kinds,  like  iron  and  time,  but  the  most  logical 
conception  is  to  classify  debits  and  credits  together  in  the 
single  class  balance.  One's  balance  is  what  he  is  worth 
when  the  debits  and  credits  have  been  compared.  If  the 
preponderance  is  on  the  side  of  debit  we  consider  the  balance 
negative,  if  on  the  side  of  credit  we  consider  the  balance 
positive.  Likewise,  we  may  consider  motion  in  each  direc- 
tion of  the  compass  as  in  a  class  by  itself,  never  using  any 
conception  of  measurement  save  the  purely  numerical  one 
of  comparing  things  which  are  exactly  of  the  same  kind 
together.  But  it  is  more  logical,  and  certainly  more  general, 
to  consider  motions  in  all  directions  of  the  compass  and 
of  any  distances  as  all  belonging  to  a  single  class  of  quantity. 

23 


24  VECTOR  CALCULUS 

In  that  case  the  comparison  of  the  different  motions  leads 
us  to  the  notion  of  complex  numbers.  When  Wessel  made 
his  study  of  the  vectors  in  a  plane  he  was  studying  the 
hypernumbers  we  usually  call  "the  complex  field."  The 
hypernumbers  had  been  studied  in  themselves  before,  but 
were  looked  upon  (rightly)  as  being  creations  of  the  mind 
and  (in  that  sense  correctly)  as  having  no  existence  in  what 
might  be  called  the  real  world.  However,  their  deduction 
from  the  vectors  in  a  plane  showed  that  they  were  present 
as  relations  of  quantities  which  could  be  considered  as  alike. 
Again  when  Steinmetz  made  use  of  them  in  the  study  of 
the  relations  of  alternating  currents  and  electromotive 
forces,  it  became  evident  that  the  so-called  power  current 
and  wattless  current  could  be  regarded  as  parts  of  a  single 
complex  current,  and  similarly  for  the  electromotive  forces. 
The  laws  of  Ohm  and  Kirchoff  could  then  be  generalized  so 
as  to  be  true  for  the  new  complex  quantities.  In  this  brief 
history  we  find  an  example  of  the  interaction  of  the  develop- 
ments of  mathematics.  The  inventions  of  mathematics 
find  instances  in  natural  phenomena,  and  in  some  cases 
furnish  new  conceptions  by  which  natural  phenomena  can 
be  regarded  as  containing  elements  that  would  ordinarily 
be  completely  overlooked. 

In  space  of  three  (or  more)  dimensions,  the  vectors 
issuing  from  a  point  in  all  directions  and  of  all  lengths 
furnish  quantities  which  may  be  considered  to  be  all  of 
the  same  kind,  on  one  basis  of  classification.  Therefore, 
they  will  define  certain  ratios  or  relations  which  may  be 
called  hypernumbers.  This  is  the  class  of  hypernumbers 
we  are  particularly  concerned  with,  though  we  shall  occa- 
sionally notice  others.  Further,  any  kind  of  quantity 
which  can  be  represented  completely  for  certain  purposes 
by  vectors  issuing  from  a  point  we  will  call  vector  quantity. 


VECTOR   FIELDS  25 

Such  quantities,  for  instance,  are  motions,  velocities, 
accelerations,  at  least  in  the  Newtonian  mechanics,  forces, 
momenta,  and  many  others.  The  object  of  VECTOR  CAL- 
CULUS is  to  study  these  hypernumbers  in  relation  to  their 
corresponding  quantities,  and  to  derive  an  algebra  capable 
of  handling  them. 

We  do  not  consider  a  vector  as  a  mere  triplex  of  ordinary  numbers. 
Indeed,  we  shall  consider  two  vectors  to  be  identical  when  they 
represent  or  can  represent  the  same  quantity,  even  though  one  is  ex- 
pressed by  a  certain  triplex,  as  ordinary  Cartesian  coordinates,  and  the 
other  by  another  triplex,  as  polar  coordinates.  The  numerical  method 
of  defining  the  vector  will  be  considered  as  incidental. 

2.  Notation.  We  shall  represent  vectors  for  the  most 
part  by  Greek  small  letters.  Occasionally,  however,  as 
in  Electricity,  it  will  be  more  convenient  to  use  the  standard 
symbols,  which  are  generally  Gothic  type.  As  indicated 
on  page  12  there  is  a  great  variety  of  notation,  and  only 
one  principle  seems  to  be  used  by  most  writers,  namely 
that  of  using  heavy  type  for  vectors,  whatever  the  style  of 
type.  In  case  the  vector  is  from  the  origin  to  the  point 
(x,  y,  z)  it  may  be  indicated  by 

Px,  y,  z> 

while  for  the  same  point  given  by  polar  coordinates  r,  <p,  6 
we  may  use 

Pr,  <p,  6) 

In  case  a  vector  is  given  by  its  components  as  X,  Y,  Z  we 
will  indicate  it  by 

?x,  y,  z 

3.  Equivalence.  All  vectors  which  have  the  same  direc- 
tion and  same  length  will  be  considered  to  be  equivalent. 
Such  vectors  are  sometimes  called  free  vectors.  The  term 
vector  will  be  used  throughout  this  book,  however,  with  no 
other  meaning. 


2G  VECTOR  CALCULUS 

In  case  vectors  are  equivalent  only  when  they  lie  on  the 
same  line,  and  have  the  same  direction  and  length,  they 
will  be  called  glissants.  A  force  applied  to  a  rigid  body 
must  be  considered  to  be  a  glissant,  not  a  vector.  In 
case  vectors  are  equivalent  only  when  they  start  at  the 
same  point  and  coincide,  they  will  be  called  radials.  The 
resultant  moment  of  a  system  of  glissants  with  respect  to  a 
point  A  is  a  radial  from  A. 

The  equivalence  of  two  vectors 

a  =  0 

implies  the  existence  of  equalities  infinite  in  number,  for 
their  projections  on  any  other  lines  will  then  be  equal.  The 
infinite  set  of  equalities,  however,  is  reducible  in  an  infinity 
of  ways  to  three  independent  equalities.  For  instance,  we 
may  write  either 
ax  =  ft.,   ay  =  fiy,   a2  =  13 z,   or  ar  =  Br,   a<p  =  ^lf>,alf!  =  /?„. 

The  equivalence  of  two  glissants  implies  sets  of  equalities 
reducible  in  every  case  to  five  independent  equalities.  The 
equivalence  of  two  radials  reduces  to  sets  of  six  equalities. 

4.  Vector  Fields.     Closely  allied  to  the  notion  of  radial 

is  that  of  vector  field.     A  vector  field  is  a  system  of  vectors 

each  associated  with  a  point  of  space,  or  a  point  of  a  surface, 

or  a  point  of  a  line  or  curve.     The  vector  is  a  function  of 

the  position  of  the  point  which  is  itself  usually  given  by  a 

vector,  as  p.     The  vector  function  may  be  monodromic  or 

polydromic.     We  will  consider  some  of  the  usual  vector 

fields. 

EXAMPLES 

(1)  Radius  Vector,  p  [L].  This  will  usually  be  indicated 
by  p.  In  case  it  is  a  function  of  a  single  parameter,  as  t, 
the  points  defined  will  lie  on  a  curve;*  in  case  it  is  a  function 

*  We  are  discussing  mainly  ordinary  functions,  not  the  "pathologic 
type." 


VECTOR   FIELDS  27 

of  two  parameters,  u,  v,  the  points  defined  will  lie  on  a 
surface.  The  term  vector  was  first  introduced  by  Hamilton 
in  this  sense.  When  we  say  that  the  field  is  p,  we  mean 
that  at  the  point  whose  vector  is  p  measured  from  the  fixed 
origin,  there  is  a  field  of  velocity,  or  force,  or  other  quantity, 
whose  value  at  the  point  is  p. 

(2)  Velocity,  a  [XT7-1].  Usually  we  will  designate  veloc- 
ity by  c.  In  the  case  of  a  moving  gas  or  cloud,  each  particle 
has  at  each  point  of  its  path  a  definite  velocity,  so  that  we 
can  describe  the  entire  configuration  of  the  moving  mass  at 
any  instant  by  stating  what  function  a  is  of  p,  that  is,  for 
the  point  at  the  end  of  the  radius  vector  p  assign  the  velocity 
vector.  The  path  of  a  moving  particle  will  be  called  a 
trajectory.  At  each  point  of  the  path  the  velocity  a  is  a 
tangent  of  the  trajectory. 

If  we  lay  off  from  a  fixed  point  the  vectors  a  which  corre- 
spond to  a  given  trajectory,  their  terminal  points  will 
lie  on  a  locus  called  by  Hamilton  the  hodograph  of  the 
trajectory.  For  instance,  the  hodographs  of  the  orbits  of 
the  planets  are  circles,  to  a  first  approximation.  If  we 
multiply  a  by  dt,  which  gives  it  the  dimensions  of  length, 
namely  an  infinitesimal  length  along  the  tangent  of  the 
trajectory,  the  differential  equation  of  the  trajectory 
becomes 

dp  =  adt. 

The  integral  of  this  in  terms  of  t  gives  the  equation  of  the 
trajectory. 

(3)  Acceleration.  t[LT~2].  An  acceleration  field  is  simi- 
lar to  a  velocity  field  except  in  dimensions.  The  accelera- 
tion is  the  rate  of  change  of  the  vector  velocity  at  a  point, 
consequently,  if  a  point  describes  the  hodograph  of  a  trajec- 
tory so  that  its  radius  vector  at  a  given  time  is  the  velocity 
in  the  trajectory  at  that  time,  the  acceleration  will  be  a 

3 


L\S  VECTOR  CALCULUS 

tangent  to  the  hodograph,  and  its  length  will  be  the  velocity 
of  the  moving  point  in  the  hodograph.  We  will  use  r  to 
indicate  acceleration. 

(4)  Momentum  Density.  T  [$QL~4].  This  is  a  vector 
function  of  points  in  space  and  of  some  number  which  can 
be  attached  to  the  point,  called  density.  In  the  case  of  a 
moving  cloud,  for  instance,  each  point  of  the  cloud  will  have 
a  velocity  and  a  density.  The  product  of  these  two  factors 
will  be  a  vector  whose  direction  is  that  of  the  velocity  and 
whose  length  is  the  product  of  the  length  of  the  velocity 
vector  and  the  density.  However,  momentum  density 
may  exist  without  matter  and  without  motion.  In  electro- 
dynamic  fields,  such  as  could  exist  in  the  very  simple  case 
of  a  single  point  charge  of  electricity  and  a  single  magnet 
pole  at  a  point,  we  also  have  at  every  point  of  space  a 
momentum  density  vector.  This  may  be  ascribed  to  the 
hypothetical  motion  of  a  hypothetical  ether,  but  the  essen- 
tial feature  is  the  existence  of  the  field.  If  we  calculate  the 
integral  of  the  projection  of  the  momentum  density  on  the 
tangent  to  a  given  curve  from  a  point  A  to  a  point  B,  the 
value  of  the  integral  is  the  action  of  an  infinitesimal  volume, 
an  action  density,  along  that  path  from  A  to  B.  The 
integration  over  a  given  volume  would  give  the  total 
action  for  all  the  particles  over  their  various  paths.  This 
would  be  a  minimum  for  the  paths  actually  described  as 
compared  with  possible  paths.  Specific  momentum  is 
momentum  density  of  a  moving  mass. 

(5)  Momentum.  Y  [TOL-1].  The  volume  integral  of 
momentum  density  or  specific  momentum  is  momentum. 
Action  is  the  line-integral  of  momentum. 

(6)  Force  Density.  F  [^QL^T-1].  If  a  field  of  momen- 
tum density  is  varying  in  time  then  at  each  point  there  is  a 
vector  which  may  be  called  force-density,  the  time  derivative 


VECTOR   FIELDS  29 

of  the  momentum  density.  Such  cases  occur  in  fields  due 
to  moving  electrons  or  in  the  action  of  a  field  of  electric 
intensity  upon  electric  density,  or  magnetic  intensity  on 
magnetic  density. 

(7)  Force.  X  [mL-1?7-1].  The  unit  of  force  has  re- 
ceived a  name,  dyne.  It  is  the  volume  integral  of  force 
density.  The  time  integral  of  a  field  of  force  is  momentum. 
In  a  stationary  field  of  force  the  line  integral  of  the  field 
for  a  given  path  is  the  difference  in  energy  between  the 
points  at  the  ends  of  the  path,  or  what  is  commonly  called 
work.  In  case  the  field  is  conservative  the  integral  has  the 
same  value  for  all  paths  (which  at  least  avoid  certain 
singular  points),  and  depends  only  on  the  end  points, 
This  takes  place  when  the  field  is  a  gradient  field  of  a  force- 
function,  or  a  potential  function.  If  we  project  the  force 
upon  the  velocity  at  each  point  where  both  fields  exist, 
the  time  integral  of  the  scalar  quantity  which  is  the  product 
of  the  intensity  of  the  force,  the  intensity  of  the  velocity 
and  the  cosine  of  the  angle  between  them,  is  the  activity  at 
the  point. 

(8)  Flux  Density.  12  [UT~1}.  In  the  case  of  the  flow  of 
an  entity  through  a  surface  the  limiting  value  of  the  amount 
that  flows  normally  across  an  infinitesimal  area  is  a  vector 
whose  direction  is  that  of  the  outward  normal  of  the  surface, 
and  whose  intensity  is  the  limit.  In  the  case  of  a  flow  not 
normal  to  the  surface  across  which  the  flux  is  to  be  de- 
termined, we  nevertheless  define  the  flux  density  as  above. 
The  flux  across  any  surface  becomes  then  the  surface 
integral  of  the  projection  of  the  flux  density  on  the  normal 
of  the  surface  across  which  the  flux  is  to  be  measured. 

Flux  density  is  an  example  of  a  vector  which  depends 
upon  an  area,  and  is  sometimes  called  a  bivector.  The 
notion  of  two  vectors  involved  in  the  term  bivector  may 


30  VECTOR  CALCULUS 

be  avoided  by  the  term  cycle,  or  the  term  feuille.  It  is 
also  called  an  axial  vector,  in  opposition  to  the  ordinary 
vectors,  called  polar  vectors.  The  term  axial  is  applicable 
in  the  sense  that  it  is  the  axis  or  normal  of  a  portion  of  a 
surface.  The  portion  (feuille,  cycle)  of  the  surface  is 
traversed  in  the  positive  direction  in  going  around  its 
boundary,  that  is,  with  the  surface  on  the  left-hand.  If 
the  direction  of  the  axial  vector  is  reversed,  we  also  traverse 
the  area  attached  in  the  reverse  direction,  so  that  in  this 
sense  the  axial  vector  may  be  regarded  as  invariant  for 
such  change  while  the  polar  vector  would  not  be  invariant. 
The  distinction  is  not  of  much  importance.  The  important 
idea  is  that  of  areal  integration  for  the  flux  density  or  any 
other  so-called  axial  vector,  while  the  polar  vector  is  sub- 
ject only  to  linear  integration.  We  meet  the  distinction 
in  the  difference  below  between  the  induction  vectors  and 
the  intensity  vectors. 

(9)  Energy  Density  Current.  R  [TOL-2?7-2].  When  an 
energy  density  has  the  idea  of  velocity  attached  to  it,  it 
becomes  a  vector  with  the  given  dimensions.  In  such 
case  we  consider  it  as  of  the  nature  of  a  flux  density. 

(10)  Energy  Current.  2  [$QT~2].  If  a  vector  of  energy 
density  current  is  multiplied  by  an  area  we  arrive  at  an 
energy  current. 

(11)  Electric  Density  Current.  J  [SL^T-1].  A  number 
of  moving  electrons  will  determine  an  average  density 
per  square  centimeter  across  the  line  of  flow,  and  the  product 
of  this  into  a  velocity  will  give  an  electric  density  current. 
To  this  must  also  be  added  the  time  rate  of  change  of 
electric  induction,  which  is  of  the  same  dimensions,  and 
counts  as  an  electric  density  current. 

(12)  Electric  Current.  C  [971-1].  The  unit  is  the  ampere 
=  3-109  e.s.  units  =  10_1  e.m.  units.  This  is  the  product 
of  an  electric  density  current  by  an  area. 


VECTOR  FIELDS  31 


(13)  Magnetic  Density  Current.  G  [$Ir2T-1}.  Though 
there  is  usually  no  meaning  to  a  moving  mass  of  magnetism, 
nevertheless,  the  time  rate  of  change  of  magnetic  induction 
must  be  considered  to  be  a  current,  similar  to  electric 
current  density. 

(14)  Magnetic  Current.  K  [^T'1].  The  unit  is  the 
heavy  side  =  1  e.m.  unit  =  3  •  1010  e.s.  units.  In  the  phenom- 
ena of  magnetic  leakage  we  have  a  real  example  of  what  may 
be  called  magnetic  current. 

Both  electric  current  and  magnetic  current  may  also  be 
scalars.  For  instance,  if  the  corresponding  flux  densities 
are  integrated  over  a  given  surface  the  resulting  scalar 
values  would  give  the  rate  at  which  the  electricity  or  the 
magnetism  is  passing  through  the  surface  per  second.  In 
such  case  the  symbols  should  be  changed  to  corresponding 
Roman  capitals. 

(15)  Electric  Intensity.  E  fMr1!1"1].  When  an  electric 
charge  is  present  in  any  portion  of  space,  there  is  at  each 
point  of  space  a  vector  of  a  field  called  the  field  of  electric 
intensity.  The  same  situation  happens  when  lines  of 
magnetic  induction  are  moving  through  space  with  a  given 
velocity.  The  electric  intensity  will  be  perpendicular  to 
both  the  line  of  magnetic  induction  and  to  the  velocity  it 
has,  and  equal  to  the  product  of  their  intensities  by  the 
sine  of  their  angle. 

The  electric  intensity  is  of  the  nature  of  a  polar  vector 
and  its  flux,  or  surface  integral  over  any  surface  has  no 
meaning.  Its  line  integral  along  any  given  path,  however, 
is  called  the  difference  of  voltage  between  the  two  points  at 
the  ends  of  the  path,  for  that  given  path.  The  unit  of 
voltage  is  the  volt  =  J  •  10~2  e.s.  units  =  108  e.m.  units. 
The  symbol  for  voltage  is  V  [$T~1].  Its  dimensions  are 
the  same  as  for  scalar  electric  potential,  or  magnetic  current. 


32  VECTOR  CALCULUS 

(16)  Electric  Induction.  D  [QL~2].  The  unit  is  the  line 
=  3-109  e.s.  units  —  10-1  e.m.  units.  This  vector  usually 
has  the  same  direction  as  electric  intensity,  but  in  non- 
isotropic  media,  such  as  crystals,  the  directions  do  not  agree. 
It  is  a  linear  function  of  the  intensity,  however,  ordinarily 

indicated  by 

D  =  k(E) 

where  k  is  the  symbol  for  a  linear  operator  which  converts 
vectors  into  vectors,  called  here  the  permittivity,  [0^>-1Z_1  T], 
measurable  in  farads  per  centimeter.  In  isotropic  media 
k  is  a  mere  numerical  multiplier  with  the  proper  dimensions, 
which  are  essential  to  the  formulae,  and  should  not  be 
neglected  even  when  k  =  1.  The  flux  is  measured  in 
coulombs. 

(17)  Magnetic  Intensity.  H  [eL"1?7"1].  The  field  due  to 
the  poles  of  permanent  magnets,  or  to  a  direct  current 
traversing  a  wire,  is  a  field  of  magnetic  intensity.  In  case 
we  have  moving  lines  of  electric  induction,  there  is  a  field  of 
magnetic  intensity.  It  is  of  a  polar  character,  and  its 
flux  through  a  surface  has  no  meaning.  The  line  integral 
between  two  points,  however,  is  called  the  gilbertage  between 
the  points  along  the  given  path,  the  unit  being  the  gilbert 
=  1  e.m.  unit  =  3  •  1010  e.s.  units.  The  symbol  is  N  [GT-1]' 
Its  dimensions  are  the  same  as  those  of  scalar  magnetic 
potential,  or  electric  current. 

(18)  Magnetic  Induction.  B  [$L~2].  The  unit  is  the 
gauss  =  1  e.m.  unit  =  3  •  1010  e.s.  units.  The  direction  is 
usually  the  same  as  that  of  the  intensity,  but  in  any  case  is 
given  by  a  linear  vector  operator  so  that  we  have 

B-m(H) 

where  \x  is  the  inductivity,  [^>0-1Z_1  T],  measurable  in  henrys 
per  centimeter.     The  flux  is  measured  in  maxwells. 


VPPf 


VECTOR   FIELDS  33 


(19)  Vector  Potential  of  Electric  Induction.  T  [eZ-1].  A 
vector  field  may  be  related  to  another  vector  field  in  a 
certain  manner  to  be  described  later,  such  that  the  first 
can  be  called  the  vector  potential  of  the  other. 

(20)  Vector  Potential  of  Magnetic  Induction.  ^  [M-1]. 
This  is  derivable  from  a  field  of  magnetic  induction.  This 
and  the  preceding  are  line-integrable. 

(21)  Hertzian  Vectors.  9,  <£.  These  are  line  integrals  of 
the  preceding  two,  and  are  of  a  vector  nature. 

5.  Vector  Lines.  If  we  start  at  a  given  point  of  a  vector 
field  and  consider  the  vector  of  the  field  at  that  point  to  be 
the  tangent  to  a  curve  passing  through  the  point,  the  field 
will  determine  a  set  of  curves  called  a  congruence,  since  there 
will  be  a  two-fold  infinity  of  curves,  which  will  at  every 
point  have  the  vector  of  the  field  as  tangent.  If  the  field 
is  represented  by  a,  a  function  of  p,  the  vector  to  a  point  of 
the  field,  then  the  differential  equation  of  these  lines  of 
the  congruence  will  be 

dp  =  adt, 

where  dt  is  a  differential  parameter.  From  this  we  can 
determine  the  equation  of  the  lines  of  the  congruence,  in- 
volving an  arbitrary  vector,  which,  however,  will  not  have 
more  than  two  essential  constants.  For  instance,  if  the 
field  is  given  by  a  =  p,  then  dp  =  pdt,  and  p  =  ael,  where 
a  is  a  constant  unit  vector.  The  lines  are,  in  this  case,  the 
rays  emanating  from  the  origin. 

The  lines  can  be  constructed  approximately  by  starting 
at  any  given  point,  thence  following  the  vector  of  the  field 
for  a  small  distance,  from  the  point  so  reached  following 
the  new  vector  of  the  field  a  small  distance,  and  so  proceed- 
ing as  far  as  necessary.  This  will  trace  approximately  a 
vector  line.  Usually  the  curves  are  unique,  for  if  the  field 
is  monodromic  at  all  points,  or  at  points  in  general,  the 


34  VECTOR  CALCULUS 

curves  must  be  uniquely  determined  as  there  will  be  at  any 
point  but  one  direction  to  follow.  Two  vector  lines  may 
evidently  be  tangent  at  some  point,  but  in  a  monodromic 
field  they  cannot  intersect,  except  at  points  where  the  in- 
tensity of  the  field  is  zero,  for  vectors  of  zero  intensity  are 
of  indeterminate  direction.  Such  points  of  intersection 
are  singular  points  of  the  field,  and  their  study  is  of  high 
importance,  not  only  mathematically  but  for  applications. 
In  the  example  above  the  origin  is  evidently  a  singular 
point,  for  at  the  origin  a  =  0,  and  its  direction  is  indetermi- 
nate. 

6.  Vector  Surfaces,  Vector  Tubes.  In  the  vector  field 
we  may  select  a  set  of  points  that  lie  upon  a  given  curve 
and  from  each  point  draw  the  vector  line.  All  such  vector 
lines  will  lie  upon  a  surface  called  a  vector  surface,  which  in 
case  the  given  curve  is  closed,  forming  a  loop,  is  further 
particularized  as  a  vector  tube.  It  is  evident  that  the  vector 
lines  are  the  characteristics  of  the  differential  equation 
dp  =  adt,  which  in  rectangular  coordinates  would  be 
equivalent  to  the  equations 

dx  _dy  _  dz 
X  ~  Y~  Z' 

In  case  these  equations  are  combined  so  as  to  give  a 
single  exact  equation,  the  integral  will  (since  it  must  con- 
tain a  single  arbitrary  constant)  be  the  equation  of  a  family 
of  vector  surfaces.  The  vector  lines  are  the  intersections 
of  two  such  families  of  vector  surfaces.  The  two  families 
may  be  chosen  of  course  in  infinitely  many  different  ways. 
Usually,  however,  as  in  Meteorology,  those  surfaces  are 
chosen  which  have  some  significance.  When  a  vector 
tube  becomes  infinitesimal  its  limit  is  a  vector  line. 

7.  Isogons.     If  we  locate  the  points  at  which  a  has  the 


VECTOR   FIELDS  35 

same  direction,  they  determine  a  locus  called  an  isogon  for 
the  field.  For  instance,  we  might  locate  on  a  weather  map 
all  the  points  which  have  the  same  direction  of  the  wind. 
If  isogons  are  constructed  in  any  way  it  becomes  a  simple 
matter  to  draw  the  vector  lines  of  the  field.  Machines  for 
the  use  of  meteorologists  intended  to  mark  the  isogons 
have  been  invented  and  are  in  use.*  As  an  instance  con- 
sider the  vector  field 

a  =  (2x,  2y,  —  z). 

An  isogon  with  the  points  at  which  a  has  the  direction  whose 
cosines  are  /,  m,  n  is  given  by  the  equations 

2x  :  2y  :   —  z  =  I :  m  :  n 
or 

2x  =  It,         2y  =  mt,        z  =  —  nt. 

It  follows  that  the  vector  to  any  point  of  this  isogon  is 
given  by 

p  =  t(l,  m,  n)  -  (0,  0,  3nt). 

That  is  to  say,  to  draw  the  vector  p  to  any  point  of  the 
isogon  we  draw  a  ray  from  the  origin  in  the  direction  given, 
then  from  its  outer  end  draw  a  parallel  to  the  Z  direction 
backward  three  times  the  length  of  the  Z  projection  of  the 
segment  of  the  ray.  The  points  so  determined  will  evi- 
dently lie  on  straight  lines  in  the  same  plane  as  the  ray  and 
its  projection  on  the  XY  plane,  with  a  negative  slope  twice 
the  positive  slope  of  the  ray.  The  tangents  of  the  vector 
lines  passing  through  the  points  of  the  isogon  will  then  be 
parallel  to  the  ray  itself.  The  vector  lines  are  drawn  ap- 
proximately by  drawing  short  segments  along  the  isogon 
parallel  to  its  corresponding  ray,  and  selecting  points  such 
that  these  short  segments  will  make  continuous  lines  in 
*Sandstrdm:  Annalen  der  Hydrographie  und  Maritimen  Meteor- 
ologie  (1909),  no.  6,  pp.  242  et.seq.  Bjerknes:  Dynamic  Meteorology. 
See  plates,  p.  50. 


36 


VECTOR   CALCULUS 


passing  to  adjacent  isogons.  The  figure  illustrates  the 
method.  All  the  vector  lines  are  found  by  rotating  the 
figure  about  the  X  axis  180°,  and  then  rotating  the  figure 
so  produced  about  the  Z  axis  through  all  angles. 


Fig.  3. 

8.  Singularities.  It  is  evident  in  the  example  preceding 
that  there  are  in  the  figure  two  lines  which  are  different 
from  the  other  vector  lines,  namely,  the  Z  axis  and  the  line 
which  is  in  the  XY  plane.  Corresponding  to  the  latter 
would  be  an  infinity  of  lines  in  the  XY  plane  passing  through 
the  origin.  These  lines  are  peculiar  in  that  the  other  vector 
lines  are  asymptotic  to  them,  while  they  are  themselves 
vector  lines  of  the  field.  A  method  of  studying  the  vector 
lines  in  the  entire  extent  of  the  plane  in  which  they  lie  was 
used  by  Poincare.     It  consists  in  placing  a  sphere  tangent 


VECTOR   FIELDS  37 

to  the  plane  at  the  origin.  Lines  are  then  drawn  from  the 
center  of  the  sphere  to  every  point  of  the  plane,  thus  giving 
two  points  on  the  sphere,  one  on  the  hemisphere  next  the 
plane  and  one  diametrically  opposite  on  the  hemisphere 
away  from  the  plane.  The  points  at  infinity  in  the  plane 
correspond  to  the  equator  or  great  circle  parallel  to  the 
plane.  In  this  representation  every  algebraic  curve  in  the 
plane  gives  a  closed  curve  or  cycle  on  the  sphere.  In  the 
present  case,  the  axes  in  the  plane  give  two  perpendicular 
great  circles  on  the  sphere,  and  the  vector  lines  will  be 
loops  tangent  to  these  great  circles  at  points  where  they 
cross  the  equator.  These  loops  will  form  in  the  four  Junes 
of  the  sphere  a  system  of  closed  curves  which  Poincare  calls 
a  topographical  system.  The  equator  evidently  belongs  to 
the  system,  being  the  limit  of  the  loops  as  they  grow  nar- 
rower. The.  two  great  circles  corresponding  to  the  axes 
also  belong  to  the  system,  being  the  limits  of  the  loops  as 
they  grow  larger.  If  a  point  describes  a  vector  line  its 
projection  on  the  sphere  will  describe  a  loop,  and  could 
never  leave  the  lune  in  which  the  projection  is  situated. 
The  points  of  tangency  are  called  nodes',  the  points  which 
represent  the  origin,  and  through  which  only  the  singular 
vector  lines  pass,  are  called  fames. 

9.  Singular  Points.  The  simplest  singular  lines  depend 
upon  the  singular  points  and  these  are  found  comparatively 
simply.     The  singular  points  occur  where 

o"  =  0         or         a  —•  oo . 

Since  we  may  multiply  the  components  of  a  by  any  ex- 
pressions and  still  have  the  lines  of  the  field  the  same,  we 
may  equally  suppose  that  the  components  of  a  are  reduced 
to  as  low  terms  as  possible  by  the  exclusion  of  common 
factors  of  all  of  them.     We  will  consider  first  the  singular 


38  VECTOR  CALCULUS 

points  for  fields  in  space,  then  those  cases  which  have 
lines  every  point  of  which  is  a  singular  point,  which  will 
include  the  cases  of  plane  fields,  since  these  latter  may  be 
considered  to  represent  the  fields  produced  by  moving  the 
plane  field  parallel  to  itself.  The  classification  given  by 
Poincare  is  as  follows. 

(1)  Node.  At  a  node  there  may  be  many  directions 
in  which  vector  lines  leave  the  point.  An  example  is  a  =  p. 
At  the  origin,  it  is  easy  to  see,  a  =  0,  and  it  is  not  possible 
to  start  at  the  origin  and  follow  any  definite  direction. 
In  fact  the  vector  lines  are  evidently  the  rays  from  the 
origin  in  all  directions.  There  is  no  other  singular  point  at 
a  finite  distance.  If,  however,  we  consider  all  the  rays  in 
any  one  plane,  and  for  this  plane  construct  the  sphere  of 
projection,  we  see  that  the  lines  correspond  to  great  circles 
on  the  sphere  which  all  pass  through  the  origin  and  the 
point  diametrically  opposite  to  it.  This  ideal  point  may 
be  considered  to  be  another  node,  so  that  all  the  vector 
lines  run  from  node  to  node,  in  this  case.  Every  vector 
line  which  does  not  terminate  in  a  node  is  a  spiral  or  a  cycle. 

(2)  Faux.  From  a  faux*  there  runs  an  infinity  of  vector 
lines  which  are  all  on  one  surface,  and  a  single  isolated 
vector  line  which  intersects  the  surface  at  the  faux.  The 
surface  is  a  singular  surface  since  every  vector  line  in  it 
through  the  faux  is  a  singular  line.  The  singular  surface 
is  approached  asymptotically  by  all  the  vector  lines  not 
singular. 

An  example  is  given  by 

a  =  (x,  y,  —  z). 

The  vector  lines  are  to  be  found  by  drawing  all  equilateral 
hyperbolas  in  the  four  quadrants  of  the  ZX  plane,  and  then 
*  Poincare  uses  the  term  col,  meaning  mountain  pass,  for  which  faux 
is  Latin. 


/ 


VECTOR    FIELDS 


39 


rotating  this  set  of  lines  about  the  Z  axis.  Evidently  all 
rays  in  the  XY  plane  from  the  origin  are  singular  lines,  as 
well  as  the  Z  axis.  Where  fauces  occur  the  singular  lines 
through  them  are  asymptotes  for  the  nonsingular  lines.     If 


Fig.  4. 

we  consider  any  plane  through  the  Z  axis,  the  system  of 
equilateral  hyperbolas  will  project  onto  its  sphere  as  cycles 
tangent  on  the  equator  to  the  great  circles  which  repre- 
sent the  singular  lines  in  that  plane.  From  this  point  of 
view  we  really  should  consider  the  two  rays  of  the  Z  axis  as 
separate  from  each  other,  so  that  the  upper  part  of  the  Z 
axis  and  the  singular  ray  perpendicular  to  it,  running  in  the 
same  general  direction  as  the  other  vector  lines,  would  con- 
stitute a  vector  line  with  a  discontinuity  of  direction,  or 
with  an  angle.  Such  a  vector  line  to  which  the  others  are 
tangent  at  points  at  infinity  only  is  a  boundary  line  in  the 
sense  that  on  one  side  we  have  infinitely  many  vector  lines 
which  form  cycles  (in  the  sense  defined)  while  on  the  other 
sides  we  have  vector  lines  which  belong  to  different  sys- 
tems of  cycles. 


40 


VECTOR  CALCULUS 


A  simple  case  of  this  example  might  arise  in  the  inward 
flow  of  air  over  a  level  plane,  with  an  ascending  motion 
which  increased  as  the  air  approached  a  given  vertical 
line,  becoming  asymptotic  to  this  vertical  line.  In  fact, 
a  small  fire  in  the  center  of  a  circular  tent  open  at  the  bottom 
for  a  small  distance  and  at  the  vertex,  would  give  a  motion 
to  the  smoke  closely  approximating  to  that  described. 

A  singular  line  from  a  faux  runs  to  a  node  or  else  is  a 
spiral  or  part  of  a  cycle  which  returns  to  the  faux. 

An  example  that  shows  both  preceding  types  is  the  field 

a  =  (x2  +  y2  —  1,  bxy  —  5,  mz). 


In  the  X  Y  plane  the  singular  points  are  at  infinity  as  follows : 
A  at  the  negative  end  of  the  X  axis,  and  B  at  the  positive 
end,  both  fauces;  C  at  the  end  of  the  ray  whose  direction 
is  tan-1  2,  in  the  first  quadrant,  D  at  the  end  of  the  ray  of 
direction  tan-1  2  in  the  third  quadrant;  E  at  the  end  of  the 


VECTOR   FIELDS  41 

ray  of  direction  tan-1  —  2  in  the  fourth  quadrant;  and  F 
at  the  end  of  the  ray  of  tan-1  —  2  in  the  second  quadrant, 
these  four  being  nodes.  Vector  lines  run  from  E  to  D 
separated  from  the  rest  of  the  plane  by  an  asymptotic 
division  line  from  B  to  D;  from  C  to  D  on  the  other  side 
of  this  division  line,  separated  from  the  third  portion  of 
the  plane  by  an  asymptotic  division  line  from  C  to  A ;  and 
from  C  to  F  in  the  third  portion  of  the  plane.  The  figure 
shows  the  typical  lines  of  the  field. 

(3)  Focus.  At  a  focus  the  vector  lines  wind  in  asymp- 
totically, either  like  spirals  wound  towards  the  vertex  of  a 
spindle  produced  by  rotating  a  curve  about  one  of  its 
tangents,  one  vector  line  passing  through  the  focus,  or 
they  are  like  spirals  wound  around  a  cone  towards  the 


Fig.  6. 
vertex.     As  an  example 

o-  =  (x+  y,  y  -  x,  z). 

The  Z  axis  is  a  single  singular  line  through  the  origin,  which 
is  a  singular  point,  a  focus  in  this  case.  The  XY  plane 
contains  vector  lines  which  are  logarithmic  spirals  wound  in 
towards   the   origin.     The   other   vector  lines   are  spirals 


42 


VECTOR  CALCULUS 


wound  on  cones  of  revolution,  their  projections  on  XY 
being  the  logarithmic  spirals.  By  changing  z  to  az  we 
would  have  different  surfaces  depending  upon  whether 

1  <  a. 


a<  1 


or 


In  case  a  spiral  winds  in  onto  a  cycle,  the  successive 
turns  approaching  the  cycle  asymptotically,  the  cycle  is 
called  a  limit  cycle.  In  this  example  the  line  at  infinity 
in  the  X  Y  plane,  or  the  corresponding  equator  on  its  sphere, 
is  a  limit  cycle.  It  is  clear  that  the  spirals  on  the  cones 
wind  outward  also  towards  the  lines  at  infinity  as  limit 
cycles.  From  this  example  it  is  plain  that  vector  lines 
which  are  spiral  may  start  asymptotically  from  a  focus  and 
be  bounded  by  a  limit  cycle.  The  limit  cycle  thus  divides 
the  plane  or  the  surface  upon  which  they  lie  into  two 
mutually  exclusive  regions.  Vector  lines  may  also  start 
from  a  limit  cycle  and  proceed  to  another  limit  cycle. 

As  an  example  of  vector  lines  of  both  kinds  consider  the 
field 


Fig.  7. 
a  =  (r2  _  1,  r2  +  lf  mz)f 

where  the  first  component  is  in  the  direction  of  a  ray  in  the 
XY  plane  from  the  origin,  the  second  perpendicular  to 


VECTOR   FIELDS 


43 


this  in  the  XY  plane,  and  the  third  is  parallel  to  the  Z  axis. 
The  vector  lines  in  the  singular  plane,  the  XY  plane,  are 
spirals  with  the  origin  as  a  focus  for  one  set,  which  wind 
around  the  focus  negatively  and  have  the  unit  circle  as  a 
limit  cycle,  while  another  set  wind  around  the  unit  circle 
in  the  opposite  direction,  having  the  line  at  infinity  as  a 
limit  cycle.     The  polar  equation  of  the  first  set  is  r~l  —  r 


An  example  with  all  the  preceding  kinds  of  singularities 
is  the  field 


Fig.  8. 

a  =  ( [r2  -  l)(r  -  9)],  (r2  -  2r  cos  9  -  8),  mz) 

with  directions  for  the  components  as  in  the  preceding 
example.  The  singular  points  are  the  origin,  a  focus;  the 
point  A  (r  =  3,  0  =  +  cos-1  §),  a  node;  the  point  B  (r  =  3, 
6  =  —  cos-1  J),  a  faux.  The  line  at  infinity  is  a  limit 
cycle,  as  well  as  the  circle  r  =  1,  which  is  also  a  vector 
line.     The  circle  r  =  3  is  a  vector  line  which  is  a  cycle, 

4 


44  VECTOR  CALCULUS 

starting  at  the  faux,  passing  through  the  node  and  returning 
to  the  faux.  The  vector  lines  are  of  three  types,  the  first 
being  spirals  that  wind  asymptotically  around  the  focus, 
out  to  the  unit  circle  as  limit  cycle;  the  second  start  at  the 
node  A  and  wind  in  on  the  unit  circle  as  limit  cycle;  the 
third  start  at  the  node  A  and  wind  out  to  the  line  at  in- 
finity as  unit  cycle.  The  second  set  dip  down  towards  the 
faux.  The  exceptional  vector  lines  are  the  line  at  infinity, 
the  unit  circle,  both  being  limit  cycles;  the  circle  of  radius 
3;  a  vector  line  which  on  the  one  side  starts  at  the  faux  B 
winding  in  on  the  unit  circle,  and  on  the  other  side  starts 
at  the  faux  B  winding  outward  to  the  line  at  infinity  as 
limit  cycle.  The  last  two  are  asymptotic  division  lines  of 
the  regions.     The  figure  exhibits  the  typical  curves. 

(4)  Faux-Focus.  This  type  of  singular  point  has  passing 
through  it  a  singular  surface  which  contains  an  infinity 
of  spirals  having  the  point  as  focus,  while  an  isolated  vector 
line  passes  through  the  point  and  the  surface.  No  other 
surfaces  through  the  vector  lines  approach  the  point.  An 
instance  is  the  field 

a-  =  (x,  y,  —  z). 

The  Z  axis  is  the  isolated  singular  line,  while  the  XY  plane 
is  the  singular  plane.  In  it  there  is  an  infinity  of  spirals 
with  the  origin  as  focus  and  the  line  at  infinity  as  limit 
cycle.  All  other  vector  lines  lie  on  the  surfaces  rz  =  const. 
These  do  not  approach  the  origin. 

(5)  Center.  At  a  center  there  is  a  vector  line  passing 
through  the  singular  point,  and  not  passing  through  this 
singular  line  there  is  a  singular  surface,  with  a  set  of  loops 
or  cycles  surrounding  the  center,  and  shrinking  upon  it. 
There  is  also  a  set  of  surfaces  surrounding  the  isolated 
singular  line  like  a  set  of  sheaths,  on  each  of  which  there  are 
vector  lines  winding  around  helically  on  it  with  a  decreasing 


VECTOR   FIELDS 


45 


Fig.  9. 


pitch  as  they  approach  the  singular  surface,  which  they 
therefore  approach  asymptotically.  As  an  instance  we 
have  the  field 

a  =  (y,  -  x,  z). 

The  Z  axis  is  the  singular  isolated  vector  line,  the  XY  plane 

the    singular    surface,  circles 

concentric  to  the    origin  the 

singular  vector  lines  in  it,  and 

the  other  vector  lines  lie  on 

circular   cylinders   about  the 

Z  axis,  approaching  the  XY 

plane  asymptotically. 

The  method  of  determining 
the  character  of  a  singular 
point  will  be  considered  later 
in  connection  with  the  study 
of  the  linear  vector  operator. 
A  singular  point  at  infinity  is  either  a  node  or  a  faux. 

10.  Singular  Lines.  Singularities  may  not  occur  alone 
but  may  be  distributed  on  lines  every  point  of  which  is  a 
singular  point.  This  will  evidently  occur  when  cr  =  0  gives 
three  surfaces  which  intersect  in  a  single  line.  The  dif- 
ferent types  may  be  arrived  at  by  considering  the  line  of 
singularities  to  be  straight,  and  the  surfaces  of  the  vector 
lines  with  the  points  of  the  singular  line  as  singularities 
to  be  planes,  -for  the  whole  problem  of  the  character  of  the 
singularities  is  a  problem  of  analysis  situs,  and  the  deforma- 
tion will  not  change  the  character.  The  types  are  then  as 
follows : 

(1)  Line  of  Nodes.  Every  point  of  the  singular  line  is  a 
node.  A  simple  example  is  a  =  (x,  y,  0).  The  vector 
lines  are  all  rays  passing  through  the  Z  axis  and  parallel 
to  the  XY  plane. 


46  VECTOR  CALCULUS 

(2)  Line  of  Fauces.  There  are  two  singular  vector 
lines  through  each  point  of  the  singular  line.  As  an  instance 
a  =  (x,  —  y,  0).  The  lines  through  the  Z  axis  parallel  to 
the  X  and  the  Y  axes  are  singular,  all  other  vector  lines 
lying  on  hyperbolic  cylinders. 

(3)  Line  of  Foci.  The  points  of  the  singular  line  are 
approached  asymptotically  by  spirals.  As  an  instance 
<t  =  (x  +  y,  y  —  x,  0).  The  vector  lines  are  logarimithic 
spirals  in  planes  parallel  to  the  XY  plane,  wound  around  the 
Z  axis  which  is  the  singular  line. 

(4)  Line  of  Centers.  A  simple  case  is  a  —  (y,  —  x,  0). 
The  vector  lines  are  the  Z  axis  and  all  circles  with  it  as  axis. 

11.  Singularities  at  Infinity.  The  character  of  these  is 
determined  by  transforming  the  components  of  a  so  as  to 
bring  the  regions  at  infinity  into  the  finite  parts  of  the 
space  we  are  considering.  The  asymptotic  lines  will  then 
have  in  the  transformed  space  nodes  at  which  the  lines  are 
tangent  to  the  asymptotic  line. 

12.  General  Characters.  The  problem  of  the  character 
of  a  vector  field  so  far  as  it  depends  upon  the  vector  lines 
and  their  singularities  is  of  great  importance.  Its  general 
resolution  is  due  to  Poincare.  In  a  series  of  memoirs  in 
the  Journal  des  Mathematiques*  he  investigated  the 
qualitative  character  of  the  curves  which  represent  the 
characteristics  of  differential  equations,  particularly  with 
the  intention  of  bringing  the  entire  set  of  integral  curves 
into  view  at  once.  Other  studies  of  differential  equations 
usually  relate  to  the  character  of  the  functions  defined  at 
single  points  and  in  their  vicinities.  The  chief  difficulty 
of  the  more  general  study  is  to  ascertain  the  limit  cycles. 
These  with  the  asymptotic  division  lines  separate  the 
field  into  independent  regions. 

*  Ser.  (3)  7  (1881),  p.  375;  ser.  (3)  8  (1882),  p.  251;  ser.  (4)  1  (1885), 
p.  167.    Also  Takeo  Wado,  Mem.  Coll.  Sci.  Tokyo,  2  (1917)  151. 


VECTOR   FIELDS  47 

The  asymptotic  division  lines  appear  on  meteorological 
maps  as  lines  on  the  surface  of  the  earth  towards  which, 
or  away  from  which,  the  air  is  moving.  They  are  called 
in  the  two  cases  lines  of  convergence,  or  lines  of  divergence, 
respectively.  If  a  division  line  of  this  type  starts  at  a 
node  the  node  may  be  a  point  of  convergence  or  a  point  of 
divergence.  The  line  will  then  have  the  same  character. 
The  node  in  other  fields,  such  as  electric  or  magnetic  or  heat 
flow,  is  a  source  or  a  sink.  If  a  division  line  starts  from  a 
faux,  the  latter  is  often  called  a  neutral  point.  A  focus  may 
be  also  a  point  of  convergence  or  point  of  divergence.  In 
the  case  of  a  singular  line  consisting  of  foci,  the  singular  line 
may  be  a  line  of  convergence  or  of  divergence;  in  the  first 
case,  for  instance,  the  singular  line  is  the  core  of  the  anti- 
cyclone, in  the  latter  case,  the  core  of  the  cyclone. 

The  limit  cycles  which  are  not  at  infinity  are  division 
lines  which  enclose  areas  that  remain  isolated  in  the  field. 
Such  phenomena  as  the  eye  of  the  cyclone  illustrate  the  oc- 
currence of  limit  cycles  in  natural  phenomena.  The  limit 
cycle  may  be  a  line  of  convergence  or  a  line  of  divergence, 
the  air  in  the  first  case  flowing  into  the  line  asymptotically 
from  both  inside  and  outside,  with  the  focus  serving  as  a 
source,  and  in  the  other  case  with  conditions  reversed. 

The  practical  handling  of  these  problems  in  meteorological 
work  depends  usually  upon  the  isogonal  lines:  the  lines 
which  are  loci  of  equidirected  tangents  of  the  vector  lines 
of  the  field.  These  are  drawn  and  the  infinitesimal  tan- 
gents drawn  across  them.  The  filling  in  of  the  vector 
lines  is  then  a  matter  of  draughtsmanship.  The  isogonal 
lines  will  themselves  have  singularities  and  these  will 
enable  one  to  determine  somewhat  the  singularities  of  the 
vector  lines  themselves.  Since  the  unit  vector  in  the 
direction  of  a  is  constant  along  an  isogon  it  is  evident  that 


48  VECTOR  CALCULUS 

the  only  change  in  a  along  an  isogon  is  in  its  intensity, 
that  is,  a  keeps  the  same  direction,  and  its  differential  is 
therefore  a  multiple  of  a,  that  is,  the  isogons  have  for  their 
differential  equation 

da  =  adt. 

Consequently,  when  a  =  0  or  a  =  <x>  the  isogon  will  have  a 
singular  point.  It  does  not  follow,  however,  that  all  the 
singular  points  of  the  isogons  will  appear  as  singular  points 
such  as  are  described  above  for  the  vector  lines.  When 
the  differential  equation  of  the  isogons  is  reduced  to  the 
standard  form 

dp  =  rdu 

we  shall  see  later  that  r  will  be  a  linear  vector  function  of  a, 
and  that  a  linear  vector  function  may  have  zero  directions, 
so  that  <pa  —  0,  without  a  =  0.  Some  of  the  phenomena 
that  may  happen  are  the  following,  from  Bjerknes'  Dynamic 
Meteorology  and  Hydrography.     See  his  plates  42a,  426. 

1.  Node  of  Isogons.  These  may  be  positive,  in  which 
case  the  directions  of  the  tangents  of  the  vector  lines  will 
increase  (that  is,  the  tangent  will  turn  positively)  as  succes- 
sive isogons  are  taken  in  a  positive  rotation  about  the  node, 
or  may  be  negative  in  the  reverse  case.  The  positive  node 
of  the  isogon  will  then  correspond  to  a  node,  a  focus,  or  a 
center  of  the  vector  lines.  The  negative  node  of  the  isogon 
will  correspond  to  a  faux  of  the  vector  lines. 

If  the  isogons  are  parallel,  having,  therefore,  a  node  at 
infinity  in  either  of  their  directions,  the  vector  lines  may 
have  asymptotic  division  lines  running  in  the  same  direc- 
tion, or  they  may  have  lines  of  inflexion  parallel  to  the 
isogons. 

2.  Center  of  Isogons.  When  the  isogons  are  cycles  they 
may  correspond  to  very  complicated  forms  of  the  vector 


VECTOR  FIELDS  40 

lines.     Several  of  these  are  to  be  found  in  a  paper  by  Sand- 

strom,  Annalen  der  Hydrographie  und  maritimen  Meteor- 

ologie,   vol.   37   (1909),   p.   242,   Uber  die  Bewegung    der 

Flussigkeiten. 

EXERCISES* 

*  To  be  solved  graphically  as  far  as  possible. 

1.  A  translation  field  is  given  by  a-  =  (at,  bt,  ct),  what  are  the  vector 
lines,  the  isogons,  and  the  singularities? 

2.  A  rotation  field  is  given  by  a  =  (mz  —  ny,  nx  —  Iz,  ly  —  mx), 
what  are  the  isogons,  singularities,  and  vector  lines? 

3.  A  field  of  deformation  proportional  to  the  distance  in  one  direction 
is  given  by  a  =  {ax,  0,  0).     Determine  the  field. 

4.  A  general  field  of  linear  deformation  is  given  by 

o-  =  (ax  +  by  +  cz,  fx  +  gy  +  hz,  kx  +  ly  +  tnz) . 
determine  the  various  kinds  of  fields  this  may  represent  according  to 
the  different  possible  cases. 

5.  Consider  the  quadratic  field* 

a  =  (x2  —  y2  —  z2,  2xy,  2xz). 

6.  Consider  the  quadratic  field  a  =  (xy  —  xz,  yz  —  yx,  zx  —  zy). 

7.  What  are  the  lines  of  flow  when  the  motion  is  stationary  in  a 
rotating  fluid  contained  in  a  cylindrical  vessel  with  vertical  axis  of 
rotation? 

8.  Consider  the  various  fields  a  =  (ay  -\-  x,  y  —  ax,  b)  for  different 
values  of  a,  which  is  the  tangent  of  the  angle  between  the  curves  and 
their  polar  radii.  What  happens  in  the  successive  diagrams  to  the 
isogons,  to  the  curves? 

9.  Consider  the  various  fieldsf  a  =  (l,f(r  —  a),  b)  where  r  is  the 
polar  radius  in  the  XY  plane,  a  is  constant,  and  /  takes  the  various 
forms 

f(x)  =  x,  x2,  x3,  x112,  x113,  x~l,  x~2,  ex,  log  x,  sin  x,  tan  x. 

10.  Consider  the  forms  a  =  (1,  f(air  sin  r),  b)  where 

j(x)  =  sin  x,  cos  x,  tan  x. 

11.  In  various  electrical  texts,  such  as  Maxwell,  Electricity  and 
Magnetism,  and  others,  there  will  be  found  plates  showing  the  lines  of 
various  fields.  Discuss  these.  Also,  the  meteorological  maps  in 
Bjerknes'  Dynamic  Meteorology,  referred  to  fibove. 

*  See  Hitchcock,  Proc.  Amer.  Acad.  Arts  and  Sci.,  12  (1917),  No.  7, 
pp.  372-454. 

f  See  Sandstrom  cited  above. 


50  VECTOR  CALCULUS 

12.  In  a  funnel-shaped,  vortex  of  a  water-spout  the  spout  may  be 
considered  to  be  made  up  of  twisted  funnels,  one  inside  another,  the 
space  between  the  surfaces  being  a  vortex  tube.  In  the  Cottage  City 
water-spout,  Aug.  19,  1896,  the  equation  of  the  outside  funnel  may  be 
taken  to  be 

(z2  +  y*)z  =  3600. 

In  this  x,  y  are  measured  horizontally  in  meters  from  the  axis  of  the 
tubes,  and  z  is  measured  vertically  downwards  from  the  cloud  base, 
which  is  1100  meters  above  the  ground.  The  inner  surfaces  have  the 
same  equation  save  that  instead  of  3600  on  the  right  we  have 
3600/(1. 60 10)2n;  that  is,  at  any  level,  the  radius  of  a  surface  bounding 
a  tube  is  found  from  the  preceding  radius  at  the  same  level  by  dividing 
by  the  number  whose  logarithm  (base  10)  is  0.20546.  From  meteoro- 
logical theory  the  velocity  of  the  wind  on  any  surface  is  given  by 

<r  =  (Cr,  Crz,  -  2Cz) 

where  the  first  component  is  the  horizontal  radial  component,  the 
second  is  the  tangential,  and  the  third  is  the  vertical  component.  C 
varies  for  the  different  surfaces,  and  is  found  by  multiplying  the  value 
for  the  outside  surface  by  the  square  of  the  number  1.6010.  In  Bige- 
low's  Atmospheric  Radiation,  etc.,  p.  200  et  seq.,  is  to  be  found  a  set  of 
tables  for  the  various  values  from  these  data  for  different  levels.  Char- 
acterize the  vortex  field  of  the  water-spout. 

13.  For  a  dumb-bell-shaped  water-spout,  likewise,  the  funnels  have 

the  equation 

(x2  +  y2)  sin  az  —  const/A 

where  A  varies  from  surface  to  surface  just  as  C  in  the  preceding 
problem.     The  velocity  is  given  by 

o-  =  (—  Aar  cos  az,  Aar  sin  az,  2A  sin  az), 

the  directions  being  horizontal  radial,  tangential  and  vertical.  For 
the  St.  Louis  tornado,  May  27,  1896,  the  following  data  are  given. 
Cloud  base  1200  meters  above  the  ground,  divided  into  121  parts 
called  degrees,  the  ground  thus  being  at  60°,  and  az  being  in  degrees. 
The  values  of  A  are  for  the  successive  funnels 

0.1573,  0.4052,  1.0437,  2.6883,  6.9247,  17.837. 

Characterize  the  vector  Ikies  of  this  vortex  field. 

14.  In  the  treatise  on  The  Sun's  Radiation,  Bigelow  gives  the  follow- 
ing data  for  a  funnel-shaped  vortex 

r2z  =  6400000/C 


W=wind--from9f 


PLATE  I 


PLATE  II 


VECTOR   FIELDS  51 

at  500  kilometers  z  =  500,  r  =  60474,  26287,  11513,  5023,  2192,  956. 

a  (Km/sec)  =  (Cr,  Crz,  -  2Cz). 
Calculate  for 

z  =  0,  500,  1000,  2000,  5000,  10000,  20000,  30000,  40000,  50000. 

The  results  of  the  calculations  give  a  vortex  field  agreeing  with  Hale's 
observations. 

The  vector  lines  in  the  last  three  problems  lie  on  the  funnel  surfaces, 
being  traced  out  in  fact  by  a  radius  rotating  about  the  axis  of  the  vortex, 
and  advancing  along  the  axis  according  to  the  law 

2d  =  -  z  +  C        for  the  funnel, 
20  =  az  +  C  for  the  dumb-bell. 

15.  Study  the  lines  on  the  plates,  which  represent  on  the  first  plate 
the  isogons  for  wind  velocities,  on  the  second  plate  the  corresponding 
characteristic  lines  of  wind  flow.  The  date  was  evening  of  Jan.  9,  1908. 
European  and  American  systems  of  numbering  directions  are  shown  in 
the  margin  of  plate  1.     See  Sandstrom's  paper  cited  above. 

13.  Congruences.  We  still  have  to  consider  the  relations 
of  the  various  vector  lines  to  each  other,  noticing  that  the 
vector  lines  constitute  geometrically  a  congruence,  that  is, 
a  two-parameter  system  of  curves  in  space.  The  con- 
sideration of  these  matters,  however,  will  have  to  be  post- 
poned to  a  later  chapter. 


CHAPTER  IV 

ADDITION  OF  VECTORS 

1 .  Sum  of  Vectors.  Geometrically,  the  sum  of  two  or 
more  vectors  is  found  by  choosing  any  one  of  them  as  the 
first,  from  the  terminal  point  of  the  first  constructing  the 
second  (any  other),  from  the  terminal  point  of  this  con- 
structing the  third  (any  of  those  left)  and  so  proceeding 
till  all  have  been  successively  joined  to  form  a  polygon  in 
space  with  the  exception  of  a  final  side.  If  now  this  last 
side  is  constructed  by  drawing  a  vector  from  the  initial 
point  of  the  first  to  the  terminal  point  of  the  last,  the  vector 
so  drawn  is  called  the  sum  of  the  several  vectors.  In 
case  the  polygon  is  already  closed  the  sum  is  a  zero  vector. 
When  the  sum  of  two  vectors  is  zero  they  are  said  to  be 
opposite,  and  subtraction  of  a  vector  consists  in  adding  its 
opposite. 

It  is  evident  from  the  definition  that  we  presuppose  a  space  in  which 
the  operations  can  be  effectively  carried  out.  For  instance,  if  the  space 
were  curved  like  a  sphere,  and  the  sum  of  two  vectors  is  found,  it  would 
evidently  be  different  according  to  which  is  chosen  as  the  first.  The 
study  of  vector  addition  in  such  higher  spaces  has,  however,  been  con- 
sidered.    Encyclopedic  des  sciences  mathematiques,  Tome  IV,  Vol.  2. 

2.  Algebraic  Sum.  In  order  to  define  the  sum  without 
reference  to  space,  it  is  necessary  to  consider  the  hyper- 
numbers  that  are  the  algebraic  representatives  of  the 
geometric  vectors.  We  must  indeed  start  with  a  given 
set  of  hypernumbers, 

which  are  the  basis  of  the  system  of  hypernumbers  we  in- 
tend to  study.  These  are  sometimes  called  imaginaries, 
because  they  are  analogous  to  V— 1.    In  the  case  of  three- 

52 


ADDITION   OF  VECTORS  53 

dimensional  space  there  are  three  such  hypernumbers  in  the 
basis.  We  combine  in  thought  a  numerical  value  with 
each  of  these,  the  field  or  domain  from  which  these  numeri- 
cal values  are  chosen  being  of  great  importance.  For  in- 
stance, we  may  limit  our  numbers  to  the  domain  of  integers, 
the  domain  of  rationals,  the  domain  of  reals,  or  to  other 
more  complicated  domains,  such  as  certain  algebraic  fields. 
We  then  consider  all  the  multiplexes  we  can  form  by  put- 
ting together  into  a  single  entity  several  of  the  hypernum- 
bers just  formed,  as  for  instance,  we  would  have  in  three- 
dimensional  space  such  a  compound  as 

p  =    («1,  7/e2,  Z€3). 

Since  we  are  now  using  the  base  hypernumbers  e  it  is  no 
longer  necessary  to  use  the  parentheses  nor  to  pay  attention 
to  the  order  of  the  terms.  We  drop  the  use  of  the  comma, 
however,  and  substitute  the  +  sign,  so  that  we  would  now 
write 

p  =   X€i  +  2/€2  +  2€3. 

We  may  now  easily  define  the  algebraic  sum  of  several 
hypernumbers  corresponding  to  vectors  by  the  formula 

Pi  =  Xi€i  +  y{€2  +  Zi*z,[    i  =  1,  2,  •  •  •  m, 

]T  Pi  =   2£i€i  +  2^€2  +  2Zi€3. 

i  =  1 

This  definition  of  course  includes  subtraction  as  a  special 
case. 

It  is  clear  from  this  definition  that  to  correspond  to  the 
geometric  definition,  it  is  necessary  that  the  units  e  corre- 
spond to  three  chosen  unit  vectors  of  the  space  under  con- 
sideration. They  need  not  be  orthogonal,  however.  The 
coefficients  of  the  e  are  then  the  oblique  or  rectangular 
coordinates  of  the  point  which  terminates  the  vector  if  it 
starts  at  the  origin. 


54  VECTOR  CALCULUS 

3.  Change  of  Basis.  We  may  define  all  the  hyper- 
numbers  of  the  system  in  terms  of  a  new  set  linearly  related 
to  the  original  set.     For  instance,  if  we  write 

€1  =   duOti  +  ai2«2  +  Ol3«3, 
€2  =   CiziOLl  +  a22«2  "T"  023«3> 

€3  =  a3iai  +  a32a2  +  «33«3, 
then  p  becomes 

P  =  (anx  +  any  +  anz)ai 

+  (aux  +  a22y  +  a32z)a2  +  (anx  -f  a2zy  +  a33z)a3. 

It  is  evident  then  that  if  we  transform  the  e's  by  a  non- 
singular  linear  homogeneous  transformation,  the  coeffi- 
cients of  the  new  basis  hypernumbers,  a,  are  the  transforms 
of  the  original  coefficients  under  the  contragredient  trans- 
formation. 

Inasmuch  as  the  transformation  is  linear,  the  transform  of 
a  sum  will  be  the  sum  of  the  transforms  of  the  terms  of  the 
original  sum.  The  transformation  as  a  geometrical  process 
is  equivalent  to  changing  the  axes.  This  process  evidently 
gives  us  a  new  triple,  but  must  be  considered  not  to  give 
us  a  new  hypernumber  nor  a  new  vector.  Indeed,  a  vector 
cannot  be  defined  by  a  triple  of  numbers  alone.  There 
is  also  either  explicitly  stated  or  else  implicitly  understood 
to  be  a  basis,  or  on  the  geometric  side  a  definite  set  of  axes 
such  that  the  triple  gives  the  components  of  the  vector 
along  these  axes.  It  is  evident  that  the  success  of  any 
system  of  vector  calculus  must  then  depend  upon  the 
choice  of  modes  of  combination  which  are  not  affected  by 
the  change  from  one  basis  to  another.  This  is  the  case 
with  addition  as  we  have  defined  it.  We  assume  that  we 
may  express  any  vector  or  hypernumber  in  terms  of  any 
basis  we  like,  and  usually  the  basis  will  not  appear. 

If  the  transformation  is  such  as  to  leave  the  angles  be- 


ADDITION   OF   VECTORS  55 

tween  ei,  e2,  e3  the  same  as  those  between  a\,  a2,  a3,  the 
second  trihedral  being  substantially  the  same  as  the  first 
rotated  into  a  new  position,  with  the  lengths  in  each  case 
remaining  units,  then  the  transformation  is  called  orthog- 
onal. We  may  define  an  orthogonal  transformation  algebra- 
ically as  one  such  that  if  followed  by  the  contragredient 
transformation  the  original  basis  is  restored. 

4.  Differential  of  a  Vector.  If  we  consider  two  points 
at  a  small  distance  apart,  the  vector  to  one  being  p,  to  the 
other  p',  and  the  vector  from  the  first  to  the  second,  Ap 
=  p'  —  p,  where  Ap  =  As-e,  e  being  a  unit  vector  in  the 
direction  of  the  difference,  we  may  then  let  one  point  ap- 
proach the  other  so  that  in  the  limit  e  takes  a  definite  posi- 
tion, say  a,  and  we  may  write  ds  for  As,  and  call  the  result 
the  differential  of  p  for  the  given  range  over  which  the  pf 
runs.  In  the  hypernumbers  we  likewise  arrive  at  a  hyper- 
number 

dp  =  dxei  -f-  dye?,  +  dzez, 

where  now  ds  is  a  linear  homogeneous  irrational  function 
of  dx,  dy,  dz,  which  =  V  (dx2  +  dy2  +  dz2)  in  case  ely  e2,  e3 
form  a  trirectangular  system  of  units. 

The  quotient  dpjdt  is  the  velocity  at  the  point  if  t  repre- 
sents the  time.  The  unit  vector  a:  is  the  unit  tangent  for 
a  curve.  We  generally  represent  the  principal  normal  and 
the  binormal  by  jS,  7  respectively.  When  p  is  given  as 
dependent  on  a  single  variable  parameter,  as  t  for  instance, 
then  the  ends  of  p  may  describe  a  curve.  We  may  have 
in  the  algebraic  form  the  coordinates  of  p  alone  dependent 
upon  the  parameter,  or  we  may  have  both  the  coordinates 
and  the  basis  dependent  upon  t.  For  instance,  we  may  ex- 
press p  in  terms  of  ei,  e2,  e3  which  are  not  dependent  upon 
t  but  represent  fixed  directions  geometrically,  or  we  may 
express  p  in  terms  of  three  hypernumbers  as  w,  r,  J*  which 


56  VECTOR  CALCULUS 

themselves  vary  with  t,  such  as  the  moving  axes  of  a  system 
in  space.  In  relativity  theories  the  latter  method  of  repre- 
sentation plays  an  important  part. 

5.  Integral  of  a  Vector.  If  we  add  together  n  vectors  and 
divide  the  result  by  n  we  have  the  mean  of  the  n  vectors, 
which  may  be  denoted  by  p.  If  we  select  an  infinite 
number  of  vectors  and  find  the  limit  of  their  sum  after 
multiplication  by  dt,  the  differential  of  the  parameter  by 
which  they  are  expressed,  such  limit  is  called  the  integral 
of  the  vector  expressed  in  terms  of  t,  and  if  we  give  t  two 
definite  values  in  the  integral  and  subtract  one  result  from 
the  other,  the  difference  is  the  integral  of  the  vector  from 
the  first  value  of  t  to  the  second.  More  generally,  if  we 
multiply  a  series  of  vectors,  infinite  in  number,  by  a  corre- 
sponding series  of  differentials,  and  find  the  limit  of  the 
sum  of  the  results,  such  limit,  when  it  exists,  is  called  the 
integral  of  the  series.  In  integration,  as  in  differentiation, 
the  usual  difficulties  met  in  analysis  may  appear,  but  as 
they  are  properly  difficulties  due  to  the  numerical  system 
and  not  to  the  hypernumbers,  we  will  suppose  that  the 
reader  is  familiar  with  the  methods  of  handling  them. 

The  mean  in  the  case  of  a  vector  which  has  an  infinite 

sequence  of  values  is  the  quotient  of  the  integral  taken  on 

some  set  of  differentials,  divided  by  the  integral  of  the  set 

of  differentials   itself.     The   examples  will   illustrate  the 

use  of  the  mean. 

EXAMPLES 

(1)  The  centroid  of  an  arc,  an  area,  or  a  volume  is  found 
by  integrating  the  vector  p  itself  multiplied  by  the  dif- 
ferential of  the  arc,  ds,  or  of  the  surface,  du,  or  of  the  volume 
dv.  The  integral  is  then  divided  by  the  length  of  the 
arc,  the  area  of  the  surface,  or  the  volume.     That  is 

-  Sheets  ffpdu  •  fffpdv  m 

P  —   — , or        - —        or        — 

b—  a  A  V 


ADDITION   OF   VECTORS 


57 


(2)  An  example  of  average  velocity  \s  found  in  the  following 
(Bjerknes,  Dynamic  Meteorology,  Part  II,  page  14)  obser- 
vations of  a  small  balloon. 


2  =  Ht.  in 

Meters 

Az 

Direction 

Velocity 

(w/sec.) 

Products 

77 
680 
960 
1240 
1530 
1810 
2090 
2430 
2730 
3040 
3400 
3710 
4030 
4400 

603 
280 
280 
290 
280 
280 
340 
300 
310 
360 
310 
320 
370 

S.  50°  E. 
S.  57°  E. 
S.  36°  E. 
S.  28°  W. 
S.    2°W. 
S.    2°W. 
S.  35°  W. 
S.  53°  W. 
S.  69°  W. 
S.  55°  W. 
S.  53°  W. 
S.  58°  W. 
S.  37°  W. 

3.4 
4.0 
5.3 
1.5 
1.8 
2.0 
1.5 
1.8 
1.8 
3.0 
2.8 
4.4 
10.2 

2050 

1120 

1484 

435 

504 

560 

510 

540 

558 

1080 

868 

1408 

3773 

To  average  the  velocities  we  notice  that  on  the  assump- 
tion that  the  upward  velocity  was  uniform  the  distances 
vertically  can  be  used  to  measure  the  time.  We  therefore 
multiply  each  velocity  by  the  difference  of  elevations 
corresponding,  the  products  being  set  in  the  last  column. 
These  numbers  are  then  taken  as  the  lengths  of  the  vectors 
whose  directions  are  given  by  the  third  column.  The 
sum  of  these  is  found  graphically,  and  divided  by  the  total 
difference  of  distance  upward,  that  is,  4323.  In  the  same 
manner  we  can  find  graphically  the  averages  for  each  1000 
meters  of  ascent.  We  may  now  make  a  new  table  in  order 
to  find  other  important  data,  as  follows : 


Height 

.     Pressure 
(ra-bars) 

Dens,  (ton/w3) 

Veloc.  . 

Spec.  Mo- 
mentum 
(ton/ra2  sec.) 

4000 

3000 

2000 

1000 

75 

622 
705 
797 
899 
1003 

0.00083 
0.00092 
0.00102 
0.00112 

3.8 
1.6 

•    2.4 
3.7 

0.0032 
0.0015 
0.0025 
0.0041 

;,s 


VECTOR   CALCULUS 


We  now  find  the  average  velocity  between  the  1000  m-bar, 
the  900  ra-bar,  the  800  m-bar,  the  700  ra-bar,  and  the  600 
ra-bar.  The  direction  is  commonly  indicated  by  the  in- 
tegers from  0  to  63  inclusive,  the  entire  circle  being  divided 
into  64  parts,  each  of  5f°.  East  is  0,  North  is  16,  NW.  is 
24,  etc.     The  following  table  is  found. 


Pressure 

Height 

Spec.  Vol. 

(m3/Ton) 

Direction 

Veloc. 

Spec.  Mo- 
mentum 

600 

700 

800 

'  900 

1000 

1002.6 

4274 

3057 

1970 

989 

99 

76 

1217 

1087 

981 

890 

890 

8 

7 

20 

25 

25 

5.2 
1.7 
2.4 
3.7 
3.4 

0.0043 
0.0016 
0.0024 
0.0042 
0.0040 

Of  course,  specific  momenta  should  be  averaged  like  veloc- 
ities but  usually  owing  to  the  rough  measurements  it  is 
sufficient  to  find  specific  momenta  from  the  average 
velocities. 


ADDITION   OF  VECTORS 


59 


EXERCISES 

1.  Average  as  above  the  following  observations  taken  at  places 
mentioned  (Bjerknes,  p.  20),  July  25,  1907,  at  7  a.m.  Greenwich  time. 


Isobar 

Dyn.  Ht. 

Az 

Direction 

Veloc. 

100 

200 

300 

400 

500 

600 

700 

800 

900 
1000 
1001.2 

16374 

11947 

9320 

7301 

5648 

4240 

3020 

1938 

975 

107 

98 

4427 

2627 

2019 

1653 

1408 

1220 

1082 

963 

867 

9 

0 

10 

18 

19 

8 

5 

4 

4 

36 

35 

4.7 
3.2 
3.4 
3.3 
2.6 
2.5 
2.5 
1.4 
4.5 
4.5 

Uccle, 
Lat.  50°  48' 
Long.  4°  22' 

100 
200 
300 
400 
500 
600 
700 
800 
900 
955.9 

16238 

11817 

9240 

7248 

5626 

4244 

3038 

1955 

977 

471 

4421 
2577 
1992 
1622 
1382 
1206 
1083 
978 
506 

3 
6 
7 
2 
3 
2 

62 
4 

30 

10.0 
6.5 
7.6 

10.2 
6.7 
6.8 
5.3 
0.6 
2.1 

Zurich, 
Lat.  47°  23' 
Long.  8°  33' 

200 
300 
400 
500 
600 
700 
800 
900 
1000 

11890 

9241 

7240 

5643 

4196 

2991 

1927 

981 

118 

17 

2649 

2001 

1597 

1447 

1205 

1064 

946 

863 

101 

59 
57 
58 
55 
49 
41 
38 
56 
55 

9.2 
10.5 
8.8 
8.0 
2.9 
2.9 
1.9 
4.3 
3.4 

Hamburg, 
Lat.  53°  33' 
Long.  9°  59' 

GO 


VECTOR   CALCULUS 


2.  If  the  direction  of  the  wind  is  registered  every  hour  how  is  the 
average  direction  found?  Find  the  average  for  the  following  observa- 
tions. 


Station 

Pikes 
Peak 

Vienna 

Mauritius 

Cordoba 

S  Orkneys 

Elev 

4308  m. 

26  m. 

15  m. 

437  m. 

25  m 

Summer 

Winter 

Dec-Feb. 

Winter 

Summer 

Time 

Vel. 

Az. 

Vel. 

Az. 

Vel. 

Az. 

Vel. 

Az. 

Vel. 

Az. 

0  a.m 

0.84 
1.34 
1.46 
1.05 
0.43 
0.66 
1.03 

100° 
83 
71 
57 
12 
279 
262 

0.47 
0.56 
0.42 
0.33 
0.22 
0.17 
0.36 

62 
61 
59 
57 
46 
303 
257 

1.00 
1.30 
1.30 
1.00 
1.10 
1.80 
2.40 

100 
97 
98 
119 
241 
312 
326 

0.94 
1.06 
1.44 
2.03 
2.17 
0.50 
2.78 

115 
111 
121 
132 
136 
252 
314 

0.52 
0.52 
0.51 
0.51 
0.52 
0.53 
0.54 

70 

2    

51 

4    

30 

6    

5 

8    

343 

10    

?m 

12  noon 

255 

14     

1.09 

256 

0.58 

242 

1.60 

332 

3.56 

315 

0.54 

245 

16     

0.95 

253 

0.64 

232 

1.30 

304 

3.36 

305 

0.54 

42 

18    

0.74 

247 

0.47 

223 

0.20 

10 

1.75 

299 

0.53 

35 

20    

0.49 

47 

0.14 

186 

0.90 

101 

0.72 

44 

0.53 

50 

22     

0.36 

153 

0.25 

72 

1.00 

102 

0.89 

128 

0.52 

60 

Bigelow,  Atmospheric  Circulation,  etc.,  pp.  313-315. 

3.  The  following  table  gives  the  mean  magnetic  deflecting  vectors, 
in  four  zones,  the  intensity  measured  in  10~6  dynes,  <p  measured  from 
S.  to  E.,  N.,  W.,  and  0  is  measured  above  the  horizon.  The  vector  is 
the  deflection  from  the  mean  position.  Find  the  average  for  each 
zone.     (Bigelow,  pp.  324-325.) 


Time 

Arctic 

N  Temperate 

Tropic 

S  Temperate 

s 

e 

<P 

s 

0 

<P 

s 

0 

<p 

s 

0 

<p 

a.m. 

0 

60 

-36° 

345° 

15 

-30° 

111° 

20 

-33° 

5° 

19 

27° 

259° 

1 

63 

-44 

355 

14 

-35 

109 

19 

-32 

16 

19 

31 

250 

2 

69 

-43 

5 

14 

-32 

102 

20 

-36 

7 

17 

35 

251 

3 

74 

-44 

16 

14 

-33 

108 

20 

-42 

6 

18 

36 

243 

4 

75 

-42 

25 

15 

-35 

112 

18 

-34 

10 

20 

36 

226 

5 

77 

-42 

30 

17 

-33 

110 

17 

-37 

6 

21 

33 

223 

6 

78 

-40 

32 

20 

-31 

112 

19 

-36 

4 

24 

31 

222 

7 

76 

-40 

36 

22 

-  6 

107 

21 

-37 

339 

26 

24 

235 

8 

65 

-37 

45 

25 

3 

99 

24 

-30 

297 

28 

28 

248 

9 

54 

-18 

68 

26 

24 

66 

26 

23 

228 

28 

33 

256 

10 

39 

31 

117 

27 

37 

49 

35 

25 

210 

26 

-27 

296 

11 

47 

44 

195 

25 

38 

312 

43 

22 

204 

25 

-37 

327 

ADDITION   OF  VECTORS  61 

4.  Find  the  resultant  attraction  at  a  point  due  to  a  segment  of  a 
straight  line  which  is  (a)  of  uniform  density,  (6)  of  density  which  varies 
as  the  square  of  the  distance  from  one  end.  What  is  the  mean  attrac- 
tion in  each  case? 

5.  Show  that  p  =  ta  +  \P$  is  the  equation  of  a  parabola,  that  the 
equation  of  the  tangent  is  p  =  Ua  +  \t\2&  +  x(a  +  ttfi),  that  tangents 
from  a  given  point  are  given  by  t  =  p  ±  V  (p2  —  2q),  the  point  being 
pa  +  q/3,  the  chord  of  contact  is  p  =  —  qP  +  y(a  +  PP)  which  has  a 
direction  independent  of  q  so  that  all  points  of  the  line  p  =  pa  +  zP 
have  corresponding  chords  of  contact  which  are  parallel.  If  a  chord 
is  to  pass  through  the  point  aa  +  bp  for  differing  values  of  p,  then 
q  =  ap  —  b  and  the  moving  point  pa  +  qP  lies  on  the  line  p  —  pa 
-f-  (ap  —  b)P,  whose  direction  is  independent  of  b. 

6.  If  a,  /S,  7  are  vectors  to  three  collinear  points,  then  we  can  find 
three  numbers  a,  b,  c  such  that 

aa  +  6/S  +  cy  =  0  =  a  +  b  +  c. 

7.  In  problem  5  show  that  if  three  points  are  taken  on  the  parabola 
corresponding  to  the  values  t\,  U,  tz,  then  the  three  points  of  intersection 
of  the  sides  of  the  triangle  they  determine  with  the  tangents  at  the 
vertices  of  the  triangle  are  collinear. 

8.  Determine  the  points  that  divide  the  segment  joining  A  and  B, 
points  with  vectors  a  and  0,  in  the  ratio  I :  m,  both  internally  and  ex- 
ternally. Apply  the  result  to  find  the  polar  of  a  point  with  respect  to 
a  given  triangle,  that  is,  the  line  which  passes  through  the  three  points 
that  are  harmonic  on'  the  three  sides  respectively  with  the  intersection 
of  a  line  through  the  given  point  and  the  vertex  opposite  the  side. 

9.  Show  how  to  find  the  resultant  field  due  to  superimposed  fields. 

10.  A  curve  on  a  surface  is  given  by  p  =  u(u,  v),  u  =  /(v),  study  the 
differential  of  p. 


CHAPTER  V 

VECTORS  IN  A  PLANE 
1.  Ratio  of  Two  Vectors.  We  purpose  in  this  chapter  to 
make  a  more  detailed  study  of  vectors  in  a  plane  and  the 
hypernumbers  corresponding.  In  the  plane  it  is  convenient 
to  take  some  assigned  unit  vector  as  a  reference  for  all 
others  in  the  plane,  though  this  is  not  at  all  necessary  in 
most  problems.  In  fact  we  go  back  for  a  moment  to  the 
fundamental  idea  underlying  the  metric  notion  of  number. 
According  to  this  a  number  is  defined  to  be  the  ratio  be- 
tween two  quantities  of  the  same  concrete  kind,  such  as 
the  ratio  of  a  rod  to  a  foot.  If  now  we  consider  the  ratio 
of  vectors,  regarding  them  as  the  same  kind  of  quantity, 
it  is  clear  that  the  ratio  will  involve  more  than  merely 
numerical  ratio  of  lengths.  The  ratio  in  this  case  is  in 
fact  what  we  have  called  a  hypernumber.  For  every  pair 
of  vectors  p,  x  there  exists  a  ratio  p  :  x  and  a  reciprocal 
ratio  x  :  p.  This  ratio  we  will  designate  by  a  roman 
character 

P 


p  :  x  =  p/x  = 


IT 


That  is  to  say,  we  may  substitute  p  for  qw. 

2.  Complex  Numbers.  If  we  draw  p  and  x  from  one 
point,  they  will  form  a  figure  which  has  two  segments  for 
sides  and  an  angle.  (In  case  they  coincide  we  still  con- 
sider they  have  an  angle,  namely  zero.)  In  this  figure  p  is 
the  initial  side  and  x  is  the  terminal  side.  Then  their 
complex  ratio  is  x  :  p.  Since  this  ratio  is  to  be  looked  upon 
as  a  multiplier,  it  is  clear  that  if  we  were  to  reduce  the 
sides  in  the  same  proportion,  the  ratio  would  not  be  changed. 

62 


VECTORS  IN  A  PLANE  63 

A  change  of  angle  would,  however,  give  a  different  ratio. 
However,  we  will  agree  that  all  ratios  are  to  be  considered 
as  equivalent,  or  as  we  shall  usually  say,  equal,  not  only 
when  the  figures  to  which  they  correspond  have  sides  in 
the  same  proportion,  but  also  when  they  have  the  same 
angles  and  sides  in  proportion,  even  if  not  placed  in  the 
plane  in  the  same  position.  For  instance,  if  the  vectors 
AB,  AC  make  a  triangle  which  is  similar  to  the  triangle 
DE,  DF,  if  we  take  the  sides  in  this  order,  then  we  shall 
consider  that  whatever  complex  or  hypernumber  multiplies 
AC  into  AB  will  also  multiply  DF  into  DE.  This  axiom 
of  equivalence  is  not  only  important  but  it  differentiates 
this  particular  hypernumber  from  others  which  might  just 
as  well  be  taken  as  fundamental.  For  instance,  the  Gibbs 
dyad  of  t  :  p  is  equally  a  hypernumber,  but  we  cannot 
substitute  for  ir  or  p  any  other  vectors  than  mere  multiples 
of  7r  or  p.  It  is  clear  that  in  the  Gibbs  dyad  we  have  a 
more  restricted  hypernumber  than  in  the  ordinary  com- 
plex number  which  has  been  just  defined,  and  which  is  a 
special  case  of  the  Hamiltonian  quaternion.  If  we  have 
a  Gibbs  dyad  q,  we  can  find  the  two  vectors  ir  and  p  save 
as  to  their  actual  lengths.  But  with  the  complex  number 
q  we  cannot  find  ir  and  p  further  than  to  say  that  for  every 
vector  there  is  another  in  the  ratio  q.  In  other  words  the 
only  transformations  allowed  in  the  Gibbs  dyad  are  transla- 
tion of  the  figure  AB,  AC  or  magnification  of  it.  In  the 
Hamiltonian  quaternion,  or  complex  number,  the  trans- 
formations of  the  figure  AB,  AC  may  be  not  only  those 
just  mentioned  but  rotation  in  the  plane. 

In  order  to  find  a  satisfactory  form  for  the  hypernumber 
q  which  we  have  characterized,  we  further  notice  that  if 
we  change  the  length  of  x  in  the  ratio  m  then  we  must 
change  q  in  the  same  ratio,  and  if  we  set  for  the  ratio  of  the 


64  VECTOR  CALCULUS 

length  or  intensity  of  w  to  that  of  p  the  number  r,  it  is  evi- 
dent that  we  ought  to  take  for  q  an  expression  of  the  form 

q  =  r<p(0), 

where  <p(6)  is  a  function  of  0,  the  angle  between  p  and  t, 
only.     Further  if  we  notice  that  we  now  have 

7T  =  r(p(6)p, 

where  the  first  factor  affects  the  change  of  length,  the 
second  the  change  of  direction,  it  is  plain  that  for  a  second 
multiplication  by  another  complex  number  q'  =  r'<p(0'), 
we  should  have 

tt'  =  r,rcp(e,)iP{e)P  =  r'rip(W  +  6)p. 

Whence  we  must  consider  that 

viO'Md)  =  <p{ef  +$)  =  view). 

These  expressions  are  functions  of  two  ordinary  numerical 
parameters,  0,  0',  and  are  subject  to  partial  differentiation, 
just  like  any  other  expressions.  Differentiating  first  as  to 
0,  then  as  to  6',  we  find  (<pf  being  the  derivative) 

<p\eM6')  =  ?'($+  ef)  =  wmb), 

whence 

.  vy)  =  V'{6')  _ 

where  &  is  a  constant  and  does  not  depend  upon  the  angle  at 
all.  It  may,  however,  depend  upon  the  plane  in  which 
the  vectors  lie,  so  that  for  different  planes  A;  may  be,  and 
in  fact  is,  different.  N 

Since,  when  0  =  0  the  hypernumber  becomes  a  mere 
numerical  multiplier, 

<p'(0)  =  MO). 

If  now  we  examine  the  particular  function 
<p(0)  =  cos  0+k  sin  6, 


VECTORS   IN  A   PLANE  65 

which  gives 

<p'(d)  =  — *  sin  $  +  k  cos  6  =  k  cos  6  +  k2  sin  6, 

we  find  all  conditions  are  satisfied  if  we  take  k2  =  —  1. 
We  may  then  properly  use  this  function  to  define  <p. 
This  very  simple  condition  then  enables  us  to  define  hyper- 
numbers  of  this  kind,  so  that  we  write 

q  =  r(cos  6  +  k  sin  9)  =  r  cks  6  =  rg, 

where  k2  =  —  1. 

3.  Imaginaries.  It  is  desirable  to  notice  carefully  here 
that  we  must  take  k2  equal  to  —  1,  the  same  negative 
number  that  we  have  always  been  using.  This  is  important 
because  there  are  other  points  of  view  from  which  the 
character  of  k  and  k2  would  be  differently  regarded.  For 
instance,  in  the  original  paper  of  Hamilton,  On  Algebraic 
Couples,  the  k,  or  its  equivalent,  is  regarded  as  a  linear 
substitution  or  operator,  which  converts  the  couple  (a,  b) 
into  the  couple  (—  b,  a).  While  it  is  true  that  we  may  so 
regard  the  imaginary,  it  becomes  at  once  obvious  that  we 
must  then  draw  distinctions  between  1  as  an  operator,  and 
1  as  a  number,  and  so  for  —  1,  and  indeed  for  any  expression 
x  +  yi.  In  fact,  such  distinctions  are  drawn,  and  we  find 
these  operators  occasionally  called  matrix  unity,  etc.  From 
the  point  of  view  of  the  hypernumber,  this  distinction  is 
not  possible.  Hypernumbers  are  extensions  of  the  number 
system,  similar  to  radicals  and  other  algebraic  numbers. 
The  fact  that,  as  we  will  see  later,  they  are  not  in  general 
commutative,  does  not  prevent  their  being  an  extension. 

4.  Real,  Imaginary,  Tensor,  Versor.  In  the  complex 
number 

q  =  r  cos  6  +  r  sin  6  •  k 

the  term  r  cos  6  is  called  the  real  part  of  q  and  may  be  written 
Rq.     The  term  r  sin  6-k  is  called  the  imaginary  part  of  q 


66  VECTOR  CALCULUS 

and  written  Iq.  The  number  r  is  called  the  tensor  of  q  and 
written  Tq.  The  expression  cos  6  +  sin  0  •  k  is  called  the 
versor  of  </  and  written  Uq.     Therefore, 

q=  Rq+  Iq=   TqUq. 

If  q  appears  in  the  form  q  —  a  +  bk  we  see  at  once  that 

Rq  =  a,       Iq=  bk,       Tq  =  V  (a2  +  b2),        6  =  taiT^/a. 

5.  Division.  If  we  have  w  =  qp,  then  we  also  write 
p  =  g-17r.     It  becomes  evident  that 

&Tl  =  RqKTqf,      Iq-'  =  -  Iql(Tq)\       Tq-*  =  1/Tq, 
Uq-1  =  cos  6  —  sin  6  •  k. 

6.  Conjugate,  Norm.  The  hypernumber  q  =  Kq  —  Rq 
—  Iq  is  called  the  conjugate  of  q.  If  q  belongs  to  the  figure 
AB,  AC,  then  q  belongs  to  an  inversely  similar  triangle,  that 
is,  a  similar  triangle  which  has  been  reflected  in  some 
straight  line  of  the  plane.  The  product  q°  =  Nq  =  (Tq)2 
is  called  the  norm  of  q.  It  also  has  the  name  modulus  of  q, 
particularly  in  the  theory  of  functions  of  complex  variables. 

Evidently, 

Rq  =  i(q  +  q),  Iq  =  h(q  -  ~q),  r1  =  W*    ^q~l  =  Uq- 

7.  Products  of  Complex  Numbers.  From  the  definitions 
it  is  clear  that  the  product  of  two  complex  numbers  q,  r, 
is  a  complex  number  s,  such  that 

Ts  =  TqTr,_        ZJ=   zq+  Zr, 
Rqr  =  Rrq  =  Rqr  =  Rrq  =  RqRr  -  Tlqlr, 
Rqr  =  Rqr  =  Rrq  =  Rrq  =  RqRr  +  Tlqlr, 
Iqr  =  Irq  =  —  Tqr  =  —  Irq  =  Rqlr  +  Rrlq, 
Iqr  =  Irq  =  —  Iqr  =  —  Irq  =  Rrlq  —  Rqlr. 

Hence  if  Rqr  =  0,  the  angles  of  q  and  r  are  complementary 
or  have  270°  for  their  sum. 


VECTORS   IN  A   PLANE  67 

If       Rqr  =  0,  the  angles  differ  by  90°.     In  particular 

we  may  take  r  =  1. 
If        Iqr  =  0,  the  angles  are  supplementary  or  opposite. 
If        Iqr  =  0,  the  angles  are  equal  or  differ  by  180°. 

8.  Continued  Products.     We  need  only  to   notice   that 


(qrs-  •  -z)  =  (z-  •  -srq). 

It  is  not  really  necessary  to  reverse  the  order  here  as  the 
products  are  commutative,  but  in  quaternions,  of  which 
these  numbers  are  particular  cases,  the  products  are  not 
usually  commutative,  and  the  order  must  be  as  here 
written. 

9.  Triangles.     If  ft  y,  5,  e  are  vectors  in  the  plane,  and 

e  =    gft  5  =   gyf 

then  the  triangle  of  ft  e  is  similar  to  that  of  y,  5,  while  if 
e  =  gft        5  =  ?7, 

the  triangles  are  inversely  similar. 

These  equations  enable  us  to  apply  complex  numbers  to 
certain  classes  of  problems  with  great  success. 

10.  Use  of  Complex  Numbers  as  Vectors.  If  a  vector  a 
is  taken  as  unit,  every  vector  in  the  plane  may  be  written 
in  the  form  qa,  for  some  properly  chosen  q.  We  may 
therefore  dispense  with  the  writing  of  the  a,  and  talk  of 
the  vector  q,  always  with  the  implied  reference  to  a  certain 
unit  a.  This  is  the  well-known  method  of  Wessel,  Argand, 
Gauss,  and  others.  However,  it  should  be  noticed  that 
we  have  no  occasion  to  talk  of  q  as  a  point  in  the  plane. 

EXAMPLES 
(1)  Calculate  the  path  of  the  steam  in  a  two-wheel  tur- 
bine from  the  following  data.    The  two  wheels  are  rigidly 
connected  and  rotate  with  a  speed  a  =  4000°  ft./sec.     Be- 


68  VECTOR  CALCULUS 

tween  them  are  stationary  buckets  which  turn  the  exhaust 
steam  from  the  buckets  of  the  first  wheel  into  those  of  the 
second  wheel.  The  friction  in  each  bucket  reduces  the  speed 
by  12%.  The  steam  issues  from  the  expansion  nozzle  at  a 
speed  of  /3  =  22002o°.  The  proper  exhaust  angles  of  the 
buckets  are  24°,  30°,  45°.  Find  the  proper  entrance  angles 
of  the  buckets. 

7  =  relative  velocity  of  steam  at  entrance  to  first  wheel. 
=  220020  -  400o  =  183024.3. 

8  =  velocity    of    issuing    steam,    88%    of    preceding, 

=    1610x56. 

€  =  entrance  velocity  to  stationary  bucket. 

=  5  +  a  =  I6IO1M  +  400o  =  1255i48.4. 
f  =  exit  -  110530. 
0  =  entrance  to  next  bucket  =  £  —  a  =  110530  —  400o 

=  78044.3. 
77  =   exit  =   69O135.     Absolute  exit  velocity  =  690i35 
+  4000  =  495ioo. 

Steinmetz,  Engineering  Mathematics. 

(2).  We  may  suppose  the  student  is  somewhat  familiar 
with  the  usual  elementary  theory  of  the  functions  of  a 
complex  variable.  If  w  is  an  analytic  function  of  z,  both 
complex  numbers,  then  the  real  part  of  w,  Rw,  considered 
as  a  function  of  x,  y  or  u,  v,  the  two  parameters  which  de- 
termine z,  will  give  a  system  of  curves  in  the  x,  y,  or  the 
u,  v  plane.  These  may  be  considered  to  be  the  transforma- 
tions of  the  curves  Rw  =  const,  which  are  straight  lines 
parallel  to  the  Y  axis  in  the  w  plane.  Similarly  for  the 
imaginary  part.     The  two  sets  will  be  orthogonal  to  each 

other,  since  the  slope  of  the  first  set  will  be ^—  /  -z —  ; 

J     *    1         1  dTIw/dTIw      _ 

and   01   the   other   set ^ —  /  —^ —  .     But  these  are 

ox   I      dy 


VECTORS   IN   A   PLANE  69 

negative  reciprocal,  since 

dRw      dTIw  dRw_        dTIw 

~      —     n  and  ~      —  ~ 

ox  oy  ay  ox 

EXERCISES 

1 .  If  a  particle  is  moving  with  the  velocity  12028°  and  enters  a  medium 
which  has  a  velocity  given  by 

<r  =  P  +  36  sin  z  [p,  0]8°, 
what  will  be  its  path? 

2.  The  wind  is  blowing  steadily  from  the  northwest  at  a  rate  of 
16  ft. /sec.  A  boat  is  carried  round  in  circles  with  a  velocity  12  ft. /sec. 
divided  by  the  distance  from  the  center.  The  two  velocities  are  com- 
pounded, find  the  motion  of  the  boat  if  it  starts  at  the  point  p  =  40°. 

3.  A  slow  stream  flows  in  at  the  point  120°  and  out  at  the  point 
12i8o°,  the  lines  of  flow  being  circles  and  the  speed  constant.  A  chip 
is  floating  on  the  stream  and  is  blown  by  the  wind  with  a  velocity 
6400.     Find  its  path. 

4.  If  a  triangle  is  made  with  the  sides  q,  r  then  R.qr  is  the  power  of 
the  vertex  with  reference  to  the  circle  whose  diameter  is  the  opposite 
side.     The  area  of  the  triangle  is  \TIqr. 

5.  The  sum  q  +  r  can  be  found  by  drawing  vectors  qa,  ra. 

6.  How  is  qr  constructed?     qr? 

7.  If  OAE  is  a  straight  line  and  OCF  another,  and  if  EC  and  AF 
intersect  in  B,  then  OA  BC  +  OC  •  AB  +  OB  •  CA  =  0.  If  0,  A,  B,  C 
are  concyclic  this  gives  Ptolemy's  theorem. 

8.  If  ABC  is  a  triangle  and  LM  a  segment,  and  if  we  construct 
LMP  similar  to  ABC,  LMQ  similar  to  BCA,  and  LMR  similar  to  CAB, 
then  PQR  is  similar  to  CAB. 

9.  If  the  variable  complex  number  u  depends  on  the  real  number  x 
as  a  variable  parameter,  by  the  linear  fractional  form 

ax  +  b 

u  - 


ex  +  d 

then  for  different  values  of  x  the  vector  representing  u  will  terminate  on 
a  circle. 

For  if  we  construct 

b 

U~d  ' 

w  = 

a 

u 

c 


70  VECTOR  CALCULUS 

this  reduces  to  —  (cx/d),  hence  the  angle  of  w,  which  is  the  angle  between 
u  —  ale  and  u  —  b/d,  is  the  angle  of  —  d/c  and  is  therefore  constant. 
Hence  the  circle  goes  through  a/c  (x  =  «)  and  b/d  (x  =  0). 

10.  If 

_  x(c  —  b)a  +  b(a  —  c)  , 
U~    k(c-b)  +  (a-  c) 

where  x  is  a  variable  real  parameter,  then  the  vector  representative  of 
u  will  terminate  on  the  circle  through  A,  B,  C,  where  OA  represents 
a,  OB  represents  6,  and  OC  represents  c. 

11.  Given  three  circles  with  centers  C1}  d,  C3,  and  O  their  radical 
center,  P  any  point  in  the  plane,  then  the  differences  of  the  powers  of 
P  with  respect  to  the  three  pairs  of  circles  are  proportional  respectively 
to  the  projections  of  the  sides  of  the  triangle  CiC2Cz  on  OP. 

12.  Construct  a  polygon  of  n  sides  when  there  is  given  a  set  of  points, 
Ci,  C2,  -  •  -,  Cn  which  divide  the  sides  in  given  ratios  ax  :  bi,  a2  :  62,  •  •  •, 
a»  :  6„. 

If  the  vertices  are  &,  £2,  •  •  • ,  in,  and  the  points  Ci,  C2,  •  •  • ,  Cn  are 
at  the  ends  of  vectors  71,  72,  •••,  yn,  we  have 

Olll  +  &lfc  =  7l(ai  +6l)       '  *  *      CLntn  +  bnh   =  7n(an  +  bn). 

The  solution  of  these  equations  will  locate  the  vertices.  When  is  the 
solution  ambiguous  or  impossible? 

13.  Construct  two  directly  similar  triangles  whose  bases  are  given 
vectors  in  the  plane,  fixed  in  position,  so  that  the  two  triangles  have  a 
common  vertex. 

14.  Construct  the  common  vertex  of  two  inversely  similar  triangles 
whose  bases  are  given. 

15.  Construct  a  triangle  ABC  when  the  lengths  of  the  sides  AB  and 
AC  are  given  and  the  length  of  the  bisector  AD. 

1G.  Construct  a  triangle  XYZ  directly  similar  to  a  given  triangle 
PQR  whose  vertices  shall  be  at  given  distances  from  a  fixed  point  0. 

Let  the  length  of  OX  be  a,  of  OY  be  6,  and  of  OZ  be  c.  Then  X  is 
anywhere  on  the  circle  of  radius  a  and  center  O.  We  have  XY/XZ 
=  PQIPR,  that  is, 

OY  -OX  =  PQ 
OZ-OX       PR' 
whence  we  have 

OXQR  +  OYRP  +  OZPQ  =  0. 
We  draw  OXK  directly  similar  to  RPQ  giving  KO/OX  =  QR/RP  and 

KO  +  OY  +  OZ  -£§  =  0,  that  is, 


VECTORS   IN   A   PLANE  71 

In  KOY  we  have  the  base  KO  and  the  length  OY  =  b,  and  length  of 

_     length  PQ 
length  RP' 

We  can  therefore  construct  KOY  and  the  problem  is  solved. 

17.  The  hydrographic  problem.  Find  a  point  X  from  which  the  three 
sides  of  a  given  triangle  ABC  are  seen  under  given  angles. 

XB/XA  =  y  cks  0,  XC/XA  =  z  cks  p. 

XB  =  XA  +  AB,  XC  =  XA  +  AC. 
Eliminate  XA  giving  2  cks  <?•# A  +  y  cksd-AC  =  BC.  Find  U  such 
that  z  AJBI7  =  Z  AXC,  Z  ACtf  =  Z  AXB,  then  BU  =  z  cks  *>. 
BA,CU  =  y  cks  OCA. 

Construct  A  ACX  directly  similar  to  A  A  UB. 

18.  Find  the  condition  that  the  three  lines  perpendicular  to  the 
three  vectors  pa,  qa,  ra  at  their  extremities  be  concurrent. 

We  have  p  +  xkp  =  q  +  ykq  =  r  +  zkr.  Taking  conjugates 
q  —  xkp  =  p  —  ykq  =  r—  zkr.  Eliminate  x,  y,  z  from  the  four 
equations. 

19.  If  a  ray  at  angle  0  is  reflected  in  a  mirror  at  angle  a  the  reflected 
ray  is  in  the  direction  whose  angle  is  2 a  —  /3.  Study  a  chain  of  mirrors. 
Show  that  the  final  direction  is  independent  of  some  of  the  angles. 

20.  Show  that  if  the  normal  to  a  line  is  a  and  a  point  P  is  distant  y 
from  the  line,  and  from  P  as  a  source  of  light  a  ray  is  reflected  from  the 
line,  its  initial  direction  being  —  qa,  then  the  reflected  ray  has  for 
equation  —  2ya  +  tqa  =  p. 

For  further  study  along  these  lines,  see  Laisant:  Theorie  et 
Application  des  Equipollences. 

11.  Alternating  Currents.  We  will  notice  an  application 
of  these  hypernumbers  to  the  theory  of  alternating  currents 
and  electromotive  forces,  due  to  C.  P.  Steinmetz. 

If  an  alternating  current  is  given  by  the  equation 

I  =  Io  cos  2wf(t  -  h), 

the  graph  of  the  current  in  terms  of  t  is  a  circle  whose 
diameter  is  70  making  an  angle  with  the  position  for  t  =  0 
of  2wfti.  The  angle  is  called  the  phase  angle  of  the  current. 
If  two  such  currents  of  the  same  frequency  are  superim- 


72  VECTOR   CALCULUS 

posed  on  the  same  circuit,  say 


we  may  set 


7  =  70  cos  2irf(t  -  ti), 
F  =  Jo'  cos  2tt/(*  -  fcO, 
sex 

70  cos  2vfh  +  h'  cos  2tt/V  =  70"  cos  2wfh, 

70  sin  2tt/<i  +  U  sin  2u//i'  =  70"  sin  2irft2t 

7"  =  70"  cos  2wf(t  -  it), 

which  also  has  for  its  graph  a  circle,  whose  diameter  is  the 
vector  sum  of  the  diameters  of  the  other  two  circles.  We 
may  then  fairly  represent  alternating  currents  of  the  simple 
type  and  of  the  same  frequency  by  the  vectors  which  are 
the  diameters  of  the  corresponding  circles.  The  same 
may  be  said  of  the  electromotive  forces. 

If  we  represent  the  current  and  the  electromotive  force 
on  the  same  diagram,  the  current  would  be  indicated  by  a 
yellow  vector  (let  us  say)  traveling  around  the  origin, 
with  its  extremity  on  its  circle,  while  at  the  same  time  the 
electromotive  force  would  be  represented  by  a  blue  vector 
traveling  with  the  same  angular  speed  around  a  circle 
with  a  diameter  of  different  length  perhaps.  The  yellow 
and  the  blue  vectors  would  generally  not  coincide,  but  they 
would  maintain  an  invariable  angle,  hence,  if  each  is  con- 
sidered to  be  represented  by  a  vector,  the  ratio  of  these 
vectors  would  be  such  that  its  angle  would  be  the  same  for 
all  times.  This  angle  is  called  the  angle  of  lag,  or  lead, 
according  as  the  E.M.F.  is  behind  the  current  or  ahead  of  it. 

The  law  connecting  the  vectors  is 

E=  ZI, 

where  E  is  the  electromotive  force  vector,  that  is,  the  vector 
diameter  of  its  circle,  7  is  the  current  vector,  the  diameter 
of  its  circle,  and  Z  is  a  hypernumber  called  the  impedance, 


VECTORS   IN  A   PLANE  73 

[<p/0],  measured  in  ohms.  The  scalar  part  of  Z  is  the 
resistance  of  the  circuit,  while  the  imaginary  part  is  the 
reactance,  the  formula  for  Z  being 

Z  =  r  —  xk. 

The  value  of  x  is  2irfL,  where/  is  the  frequency,  [T~l],  and 
L  is  the  inductance,  [^G-1?1],  in  henry s,  or  —  l/2irfC  where 
C  is  the  permittance,  [OT1^-1],  in  farads.  [1  farad  =  9- 1011 
e.s.  units  =  10-9  e.m.  units,  and  1  ^nn/  =  ^lO-11  e.s. 
units  =  109  e.m.  units.]  It  is  to  be  noticed  that  reactance 
due  to  the  capacity  of  the  circuit  is  opposite  in  sign  to 
that  due  to  inductance. 

The  law  above  is  called  the  generalized  Ohm's  law.  We 
may  also  generalize  KirchofFs  laws,  the  two  generalizations 
being  due  to  Steinmetz,  and  having  the  highest  importance, 
inasmuch  as  by  the  use  of  these  hypernumbers  the  same 
type  of  calculation  may  be  used  on  alternating  circuits  as 
on  direct  circuits.  The  generalization  of  KirchofFs  laws 
is  as  follows : 

(1)  The  vector  sum  of  all  electromotive  forces  acting  in  a 
closed  circuit  is  zero,  if  resistance  and  reactance  electro- 
motive forces  are  counted  as  counter  electromotive  forces. 

(2)  The  vector  sum  of  all  currents  directed  toward  a 
distributing  point  is  zero. 

(3)  In  a  number  of  impedances  in  series  the  joint  im- 
pedance is  the  vector  sum  of  all  the  impedances,  but  in  a 
parallel  connected  circuit  the  joint  admittance  (reciprocal 
of  impedance)  is  the  sum  of  the  several  admittances. 

The  impedance  gives  the  angle  of  lag  or  lead,  as  the  angle 
of  a  hyper  number  of  this  type. 

We  desire  to  emphasize  the  fact  that  in  impedances  we 
have  physical  cases  of  complex  numbers.  They  involve 
complex  numbers  just  as  much  as  velocities  involve  positive 


74  VECTOR  CALCULUS 

of  negative  velocity,  or  rotations  involve  positive  or  nega- 
tive. We  may  also  affirm  that  the  complex  currents  and 
electromotive  forces  are  real  physical  existences,  every 
current  implying  a  power  current  and  a  wattless  current 
whose  values  lag  90°  (as  time)  behind  the  power  current. 
The  power  electromotive  force  is  merely  the  real  part  of 
the  complex  electromotive  force,  and  the  wattless  E.M.F.  the 
imaginary  part  of  the  complex  electromotive  force,  both 
being  given  by  the  complex  current  and  the  complex 
impedance. 

We  find  at  the  different  points  of  a  transmission  line  that 
the  complex  current  and  complex  electromotive  force  satisfy 
the  differential  equations 

dl/ds  =  (g  +  Cok)E,        dE/ds  =  (r  +  Look)L 

The  letters  stand  for  quantities  as  follows:  g  is  mhos  I  mile, 
r  is  ohms/mile,  C  is  farads/mile,  L  is  henrys/mile.     co  =  2irf. 
Setting 

m*  =  (r  +  Lo>k)(g  +  Cirk),     I2  =  (r  +  Lak)/(g  +  Cwk), 
so  that  m  is  [X-1]  while  /  is  ohms/mile,  the  solution  of  the 
equations  is 

E  =  E0  cosh  ms  +  ll0  sinh  ms, 
I  =  Iq  cosh  ms  +  1~1Eq  sinh  ms, 

where  E0  and  70  are  the  initial  values,  that  is,  where  s  =  0. 
If  we  set  Eq  =  ZqIq  and  then  set  Z0  =  Z  cosh  h,  I  = 
Z  sinh  h  we  have 

E  =  Z  cosh  (ms  +  h)I0,        I  =  l~lZ  sinh  (ms  +  h)I0, 

E  =  I  coth  (ms  +  h)I, 
E  =  sech  h  cosh  (ms  +  h)E0, 
I  =  csch  h  sinh  (ms  +  h)I0. 

To  find  where  the  wattless  current  of  the  initial  station  has 
become  the  power  current  we  set  I  =  kl0,  that  is, 

sinh  (ms  -f-  h)  =  k  sinh  h. 


VECTORS   IN   A   PLANE  75 

The  value  of  s  must  be  real. 

EXAMPLES 
(1)  Let  r  =  2  ohms/mile,  L  =  0.02  henrys/mile, 

C  =  0.0000005  farads/mile, 
g  =  0,  to  =  2000,  coL  =  40  ohms/mile,  conductor 

reactance, 
r  +  Look  =  2  +  40/c  ohms/mile  impedance 

=  40.587.i5o. 
uC  =  0.001  mhos/mile  dielectric  susceptance. 
g  +  Coik  =  0.001  k    mhos/mile    dielectric    admit- 
tance =  0.00190°. 
(g  +  Cuk)~l  =  1000/j"1  =  100027o°  ohms/mile 

dielectric  impedance. 
m2  =  0.0405i77.i5°,         m  =  0.200188.58°, 
P  =  40500_.2.85°,  I  =  201.25_i.43°. 

Let  the  values  at  the  receiver  (s  =  0)  be 

E0  =  10000o  volts,  70  =  00o. 
Then  we  have       E  =  1000  cosh  s0.200188.58°, 

for  s  =  100     E  =  1000  cosh  20.0188.58  =  625.945.92°, 

I  =  2.7727o, 
for  s  =  8         E  =  50.01i26.ot, 
for  s  =  16       E  =  1001i80.3°, 
for  s  =  15.7    E  =  1000i8o°,  a  reversal  of  phase, 
for  s  =  7.85    E  =  090o. 

At  points  distant  31.4  miles  the  values  are  the  same. 
If  we  assume  that  at  the  receiver  end  a  current  is  to  be 
maintained  with 

Jo  =  5040°        with        E0  =  10000o, 

E  =  1000  cosh  s0.200188.58°  +  1006238.57°  sinh  s0.200188.58°, 
I  =  504o°  cosh  sm  +  5i.43°  sinh  sm. 
At  s  =  100  E  =  10730n355°. 

MacMahon,  Hyperbolic  Functions. 


76  VECTOR  CALCULUS 

(2)  Let  E0  -  10000,   70  -  65i3.5°   r  =  1,  g  =  0.00002 

Ceo  =  0.00020    period  221.5  miles,  o>L  =  4. 

(3)  The  product  P  =  EI  represents  the  power  of  the 
alternating  current,  with  the  understanding  that  the  fre- 
quency is  doubled.  The  real  or  scalar  part  is  the  effective 
power,  the  imaginary  part  the  wattless  or  reactive  power. 
The  value  of  TP  is  the  total  apparent  power.  The  cos  z  P 
is  the  power  factor,  and  sin  /  P  is  the  induction  factor. 
The  torque,  which  is  the  product  of  the  magnetic  flux  by 
the  armature  magnetomotive  force  times  the  sine  of  their 
angle  is  proportional  to  TIP,  where  E  is  the  generated 
electromotive  force,  and/  is  the  secondary  current.  In 
fact,  the  torque  is  TI'EI-p/2irf  where  p  is  the  number  of 
poles  (pairs)  of  the  motor. 

12.  Divergence  and  Curl.  In  a  general  vector  field  the 
lines  have  relations  to  one  another,  besides  having  the 
peculiarities  of  the  singularities  of  the  field.  The  most 
important  of  these  relations  depend  upon  the  way  the  lines 
approach  one  another,  and  the  shape  and  position  of  a 
moving  cross-section  of  a  vector  tube.  There  is  also  at 
each  point  of  the  field  an  intensity  of  the  field  as  well  as  a 
direction,  and  this  will  change  from  point  to  point. 

Divergence  of  Plane  Lines.  If  we  examine  the  drawing 
of  the  field  of  a  vector  distribution  in  a  plane,  we  may 
easily  measure  the  rate  of  approach  of  neighboring  lines. 
Starting  from  two  points,  one  on  each  line,  at  the  intersec- 
tion of  the  normal  at  a  point  of  the  first  line  and  the  second 
line,  we  follow  the  two  lines  measuring  the  distance  apart 
on  a  normal  from  the  first.  The  rate  of  increase  of  this 
normal  distance  divided  by  the  normal  distance  and  the 
distance  traveled  from  the  initial  point  is  the  divergence  of 
the  lines,  or  as  we  shall  say  briefly  the  geometric  divergence 
of  the  field.     It  is  easily  seen  that  in  this  case  of  a  plane 


VECTORS   IN  A   PLANE  77 

field  it  is  merely  the  curvature  of  the  curves  orthogonal 
to  the  curves  of  the  field. 

For  instance,  in  the  figure,  the  tangent  to  a  curve  of  the 
field  is  a,  the  normal  at  the  same  point  /5.  The  neighboring 
curve  goes  through  C.  The  differential  of  the  normal, 
which  is  the  difference  of  BD  and 
AC,  divided  by  AC,  or  BD,  is  the 
rate  of  divergence  of  the  second  curve 
from  the  first  for  the  distance  AB. 
Hence,  if  we  also  divide  by  AB  we 
will  have  the  rate  of  angular  turn  of 
the  tangent  a  in  moving  to  the  neigh- 
boring curve,  the  one  from  C.  This  rate  of  angular  turn 
of  the  tangent  of  the  field  is  the  same  as  the  rate  of  turn  of 
the  normal  of  the  orthogonal  system,  and  is  thus  the  curva- 
ture of  the  normal  system. 

Curl  of  Plane  Lines.  If  we  find  the  curvature  of  the 
original  lines  of  the  field  we  have  a  quantity  of  much  im- 
portance, which  may  be  called  the  geometric  curl.  This 
must  be  taken  plus  when  the  normal  to  the  field  on  the 
convex  side  of  the  curve  makes  a  positive  right  angle  with 
the  tangent,  and  negative  when  it  makes  a  negative  right 
angle  with  the  tangent.  Curl  is  really  a  vector,  but  for 
the  case  of  a  plane  field  the  direction  would  be  perpendicular 
to  the  plane  for  the  curl  at  every  point,  and  we  may  con- 
sider only  its  intensity. 

Divergence  of  Field.  Since  the  field  has  an  intensity  as 
well  as  a  direction,  let  the  vector  characterizing  the  field 
be  cr  =  Ta-a.  Then  the  rate  of  change  of  TV  in  the  direc- 
tion of  a,  the  tangent,  is  represented  by  daT<r.  Let  us 
now  consider  an  elementary  area  between  two  neighboring 
curves  of  the  field,  and  two  neighboring  normals.  If  we 
consider  Ta  as  an  intensity  of  some  quantity  whose  amount 


78  VECTOR  CALCULUS 

depends  also  upon  the  length  of  the  infinitesimal  normal 
curve,  so  that  we  consider  the  quantity  Ta-dn,  then  the 
value  of  this  quantity,  which  we  will  call  the  transport  of 
the  differential  tube  (strip  in  the  case  of  a  plane  field), 
TV  being  the  density  of  transport,  will  vary  for  different 
cross-sections  of  the  tube,  and  for  the  case  under  considera- 
tion, would  be  Ta'dn'  -  Tadn.  But  TV'  =  TV  +  daTa-ds 
and  dn'  =  dn  +  ds-dn  times  the  divergence  of  the  lines. 
Therefore,  the  differential  of  the  transport  will 
P"     ~T~(    be  (to  terms  of  the  first  order)  ds  X  dn  X  ( TV 

I L—    times  divergence  +  daTa).    Hence,  the  density 

F  '  of  this  rate  of  change  of  the  transport  is  TV 
.  times  the  divergence  +  the  rate  of  change  of  TV 
along  the  tangent  of  the  vector  line  of  the  field.  This  quan- 
tity we  call  the  divergence  of  the  field  at  the  initial  point,  and 
sometimes  it  will  be  indicated  by  div.  cr,  sometimes  by 
—  SVa,  a  notation  which  will  be  explained.  It  is  clear 
that  if  the  lines  of  a  field  are  perpendicular  to  a  set  of  straight 
lines,  since  the  curvature  of  the  straight  lines  is  zero,  the 
divergence  of  the  original  lines  is  zero,  and  the  expression 
reduces  to  daT<r. 

Curl  of  Field.  We  may  also  study  the  circulation  of  the 
vector  a  along  its  lines,  by  which  we  mean  the  product  of 
the  intensity  TV  by  a  differential  arc,  that  is,  Tads.  On 
the  neighboring  vector  line  there  is  a  different  intensity, 
TV',  and  a  different  differential  arc  ds'.  The  differential 
of  the  circulation  is  easily  found  in  the  same  manner  as 
the  divergence,  and  turns  out  to  be 

—  (dfiTa  +  TV  X  curl  of  the  vector  lines). 

This  quantity  we  shall  call  the  curl  of  the  field,  written 
sometimes  curl  a,  and  more  frequently  Wa,  which  notation 
will  be  explained. 


1 


VECTORS   IN  A   PLANE  79 

It  is  evident  that  the  curl  of  a  is  the  line  integral  of  the 
Tads  around  the  elementary  area,  for  the  parts  contributed 
by  the  boundary  normal  to  the  field  will  be  zero.  Hence, 
we  may  say  that  curl  a  is  the  limit  of  the  circulation  of  <r 
around  an  elementary  area  constructed  as  above,  to  the 
area  enclosed.  We  will  see  later  that  the  shape  of  the  area 
is  not  material. 

Likewise,  the  divergence  is  clearly  the  ratio  to  the  elemen- 
tary area  of  the  line  integral  of  the  normal  component  of  <r 
along  the  path  of  integration.  We  will  see  that  this  also 
is  independent  of  the  shape  of  the  area. 

Further,  we  see  that  in  a  field  in  which  the  intensity  of  a 
is  constant  the  divergence  becomes  the  geometric  divergence 
times  the  intensity  TV,  and  the  curl  becomes  the  geometric 
curl  times  the  intensity  T<r. 

Divergence  and  curl  have  many  applications  in  vector 
analysis  in  its  applications  to  geometry  and  physics.  These 
appear  particularly  in  the  applications  to  space.  A  simple 
example  of  convergence  or  divergence  is  shown  in  the 
changing  density  of  a  gas  moving  over  a  plane.  A  simple 
caSfc  of  curl  is  shown  by  a  needle  imbedded  in  a  moving 
viscous  fluid.  The  angular  rate  of  turn  of  the  direction  of 
the  needle  is  one-half  the  curl  of  the  velocity. 

13.  Lines  as  Levels.  If  the  general  equation  of  a  given 
set  of  curves  is 

u(x,  y)  =  c, 

the§e  curves  will  be  the  vector  lines  of  an  infinity  of  fields, 
for  if  the  differential  equation  of  the  lines  is 

dx/X  m  dy/Y, 
then  we  must  have 

Xdu/dx  +  Ydu/dy  =  0 
and  for  the  field 

a  =  Xa  +  Y0. 


80  VECTOR  CALCULUS 

We  may  evidently  choose  X  arbitrarily  and  then  find  Y 
uniquely  from  the  equation.  However,  if  a\  is  any  one 
field  so  determined,  any  other  field  is  of  the  form 

a  =  <TiR(x,  y). 

The- orthogonal  set  of  curves  would  have  for  their  finite 
equation 

v(x,  y)  =  c 

and  for  their  differential  equation 

Xdvldy  -  Ydv/dx  =  0. 

If  we  use  a  uniformly  to  represent  the  unit  tangent  of 
the  u  set,  and  P  the  unit  tangent  of  the  v  set,  then  P  =  ha. 
The  gradient  of  the  function  u  is  then  d0u-(3,  and  the 
gradient  of  the  function  v  is  —  dav-a.  But  the  gradient 
of  u  is  also  (ux,  uy)  and  of  v  is  (vx,  vv)  =  (uVf  —  ux).  It 
follows  that  the  tensors  of  the  gradients  are  equal.  In  fact, 
writing  Vm  for  gradient  u,  we  have  Vt>  =  kVu.  We  also 
have  for  whatever  fields  belong  to  the  two  sets  of  orthog- 
onal lines  for  u  curves,  a  =  rVv,  for  the  v  curves,  a'  =  sVu, 
or  also  we  may  write 

Vv  =  tot,         Vu  =  tp,        a  =  Ta-ct. 

14.  Nabla.  The  symbol  V  is  called  nabla,  and  evidently 
may  be  written  in  the  form  ad/dx  +  Pd/dy  for  vectors  in 
a  plane.  We  will  see  later  that  for  vectors  in  space  it 
may  be  written  ad/dx  +  Pd/dy  +  yd/dz,  where  a,  ft  y  are 
the  usual  unit  vectors  of  three  mutually  perpendicular 
directions.  However,  this  form  of  this  very  important 
differential  operator  is  not  at  all  a  necessary  form.  In 
fact,  if  a  and  fi  are  any  two  perpendicular  unit  vectors  in 
a  plane,  and  dr,  ds  are  the  corresponding  differential  dis- 
tances in  these  two  directions,  then  we  have 

V  =  ad/dr  +  pd/ds. 


VECTORS   IN  A   PLANE  81 

For  instance,  if  functions  are  given  in  terms  of  r,  6,  the 
usual  polar  coordinates,  then  V  =  Upd/dr  +  kUpd/rdd. 
The  proof  that  for  any  orthogonal  set  of  curves  a  similar 
form  is  possible,  is  left  to  the  student.  In  general,  V  is 
defined  as  follows :  V  is  a  linear  differentiating  vector 
operator  connected  with  the  variable  vector  p  as  follows: 
Consider  first,  a  scalar  function  of  p,  say  F(p).  Differentiate 
this  by  giving  p  any  arbitrary  differential  dp.  The  result 
is  linear  in  dp,  and  may  be  looked  upon  as  the  product  of 
the  length  of  dp  and  the  projection  upon  the  direction  of 
dp  of  a  certain  vector  for  each  direction  dp.  If  now  these 
vectors  so  projected  can  be  reduced  to  a  single  vector, 
this  is  by  definition  VF.  For  instance,  if  F  is  the  distance 
from  the  origin,  then  the  differential  of  F  in  any  direction 
is  the  projection  of  dr  in  a  radial  direction  upon  the  direc- 
tion of  differentiation.  Hence,  V7p  =  Up.  In  the  case 
of  plane  vectors,  VF  will  lie  in  the  plane.  In  case  the 
differential  of  F  is  polydromic,  we  define  VF  as  a  poly- 
dromic  vector,  which  amounts  to  saying  that  a  given  set 
of  vectors  will  each  furnish  its  own  differential  value  of  dF. 
In  some  particular  regions,  or  at  certain  points,  the  value 
of  J7F  may  become  indefinite  in  direction  because  the 
differentials  in  all  directions  vanish.  Of  course,  functions 
can  be  defined  which  would  require  careful  investigation 
as  to  their  differentiability,  but  we  shall  not  be  concerned 
with  such  in  this  work,  and  for  their  adequate  treatment 
reference  is  made  to  the  standard  works  on  analysis. 

We  must  consider  next  the  meaning  of  V  as  applied  to 
vectors.  It  is  evident  that  if  V  is  to  be  a  linear  and  there- 
fore distributive  operator,  then  such  an  expression  as  Va 
must  have  the  same  meaning  as  VXa.  +  V  Y(3  +  VZy  if 
a  =  Xa  +  F/3  -r  Zy,  where  a,  0,  y  are  any  independent 
constant  vectors.     This  serves  then  as  the  definition  of 


82  VECTOR   CALCULUS 

Vo-,  the  only  remaining  necessary  part  of  the  definition  is 
the  vector  part  which  defines  the  product  of  two  vectors. 
This  will  be  considered  as  we  proceed. 

15.  Nabla  as  a  Complex  Number.  We  will  consider  now 
p  to  represent  the  complex  number  x  +  yk,  or  re,  and  that 
all  our  expressions  are  complex  numbers.  The  proper 
expression  for  V  becomes  then 

V  =  d/dx  +  kd/dy  =  Upd/dr  +  kUpd/rdd. 

In  general  for  the  plane,  let  p  depend  upon  two  parameters 
u,  v,  and  let 

dp  =  p\du  -f-  p2dv. 

If  a  is  a  function  of  p  (generally  not  analytic  in  the  usual 
sense)  and  thus  dependent  on  u,  v,  we  will  have 

da  =  dcr/du-du  +  da/dv-dv  =  R-dpV  -a. 

If  we  multiply  dp  by  kpi,  which  is  perpendicular  to  pi,  the 
real  part  of  both  sides  will  be  equal  and  we  have,  since  kpi 
is  perpendicular  to  pi, 

Rkpidp  —  dvRkpip2, 
and  similarly 

Rkpidp  =  duRkpipi  =  —  duRkpip2 

since  the  imaginary  part  of  pip2  equals  —  the  imaginary  part 
of  p2pi- 

Substituting  in  da  we  have 

A,  =  «.*,(-,*£- £+#-£)  <r. 

\      Rkpip2ou      Rkpip2dvJ 

The  expression  in  (),  however,  is  exactly  what  we  have  de- 
fined above  as  V,  and  thus  we  have  proved  that  we  may 
write  V  in  the  form  corresponding  to  dp  in  terms  of  u  and  v : 

V  =  k(p2d/du  —  pid/dv)/Rkpip2. 

In  case  pi  and  p2  are  perpendicular  the  divisor  evidently 


VECTORS   IN  A   PLANE  83 

reduces  to  ±  Tp\Tp2  according  as p2  is  negatively  perpendic- 
ular to  pi  or  positively  perpendicular  to  it.  We  may  write 
V  in  this  case  in  the  form  (since  p2  =  —  kpi-  Tp2/Tpi  or 
+  kpr  Tp2/Tpi) 

v  =  _pi_A  .  _Pi_ A  =  pfii,p-i 1_ . 

TPl2du^  TP22dv     F      du^Ht      dv 

In  any  case  we  have  dF  =  Rdp\/F,  da  =  Rdp\7  -v. 

Also  in  any  case  V  =  Vu-d/du  +  \7v-d/dv. 

16.  Curl,  Divergence,  and  Nabla.  Suppose  now  that  a 
is  the  complex  number  for  the  unit  tangent  of  one  of  a  set 
of  vector  lines,  and  |8  the  complex  number  for  the  unit 
tangent  of  the  orthogonal  set,  at  the  same  point.  The 
curvature  of  the  orthogonal  set  is  the  intensity  of  the  vector 
rate  of  change  of  (3  along  the  orthogonal  curve.  But  this 
is  the  same  as  the  rate  of  change  of  the  unit  tangent  a  as 
we  pass  along  the  orthogonal  curve  from  one  vector  line  to 
an  adjacent  one.  The  differential  of  a  is  perpendicular 
to  a,  and  hence  parallel  to  the  direction  of  /3.  Hence  this 
curvature  can  be  written 

But  if  we  also  consider  the  value  of  R-  a(R-aV)a,  since  the 
differential  of  a  in  the  direction  of  a  has  no  component 
parallel  to  a,  this  term  is  zero,  and  may  be  added  to  the 
preceding  without  affecting  its  value.  Hence  the  curvature 
of  the  orthogonal  set  reduces  to 

R(aRaV  +  ^/3V)«  =  R-Va. 

This  is  the  divergence  of  the  curves  of  a. 

If  now  <j  =  Tcr-a,  we  find  from  the  definition  of  the 
divergence  of  a  that  it  is  merely 

R-Va. 
Considering  in  the  same  manner  the  definition  of  curl  of  a, 


84  VECTOR  CALCULUS 

we  find  it  reduces  to  —  R-kV<r,  and  if  we  multiply  this  by  k, 
so  that  we  have 


curl  a  =  -  kRW(T=LV<rf 

we  see  at  once  that  when  added  to  the  expression  for  the 
divergence  of  a  we  have 

div-<7  +  curl  <r  =  V<r. 

The  real  part  of  this  expression  is  therefore  the  divergence 
of  a,  and  the  imaginary  part  is  the  curl  of  a.  This  will 
agree  with  expressions  for  curl  and  divergence  for  space  of 
three  dimensions.  We  have  thus  found  some  of  the 
remarkable  properties  of  the  operator  V . 

17.  Solenoidal  and  Lamellar  Vector  Fields.  When  the 
divergence  of  a  is  everywhere  zero,  the  field  is  said  to  be 
solenoidal.  If  the  curl  is  everywhere  zero,  the  field  is  called 
lamellar. 

18.  Properties  of  the  Field.  Let  a  set  of  curves  u  =  c  be 
considered,  and  the  orthogonal  set  v  —  a,  and  let  the  field  a 
be  expressed  in  the  form 

o-  =  XVu  +  FVfl, 

where  it  is  assumed  that  the  gradients  Vu,  Vv  exist  at  all 
points  to  be  considered.     We  have  then 

diver  =  RVa  =  RvXVu+  RvYVv  _ 

+  XRWu+  YRWv. 

The  expression  RWu  is  called  the  plane  dissipation  of  u. 
In  case  it  vanishes  it  is  evident  that  u  satisfies  Laplace's 
equation,  and  is  therefore  harmonic. 
We  also  have 

curl  o-  =  I-  V<r  =  —  kRkVXVu  —  kRkvYVv, 

the  other  parts  vanishing. 


VECTORS   IN   A   PLANE  85 

Since  we  have  chosen  orthogonal  sets  of  curves  we  may 
write  these  in  the  forms 

diver  =  (TVu)2dX/du  +  (TVv)2dY/dv 

+  XRVVu  +  YRvVv, 
curl  o-  =  (TVu)(TW)(dY/du  -  dX/dv)k. 

In  case  we  have  chosen  the  lines  of  cr  for  the  u  curves, 
then  X  =  0,  and  a  =  Y  V  v 

diver  =  YRVW+  (TVv)2dY/dv, 
curl  (7=  TVuTVvdY/du-k. 

We  notice  that  curl  Vu  =  0,  curl  Vv  =  0,  div  k\/u  —  0, 
divkVv  =  0,  kVu  =  VvTVu/TVv,  and  for 

Y  =  TVu/TVv, 
we  have 

(TVu)-2RvS7u  =  d  log  (TVu/TVv)/du, 
'  (fV*)~*BVV«  =  d  log  (TVv/TVu)/dv. 

We  may  now  draw  some  conclusions  as  to  the  types  of 
curves  and  <r.  (Cf.  B.  O.  Peirce,  Proc.  Amer.  Acad.  Arts 
and  Sci.,  38  (1903)  663-678;  39  (1903)  295-304.) 

(1)  The  field  will  be  solenoidal  if  diver  =  0,  hence 

d  log  Y/dv  =  -  RVW/TW2,      ' 
which  may  be  integrated,  giving 

Y  =  ef(u'  v)  +  o{u). 

If  v  is  harmonic,  Y  is  a  function  of  u  only  and  a  =G(u)Vv. 

(2)  If  the  field  is  lamellar,  curl  a  =  0,  and  Y  is  a  function 
of  v  only,  so  that  a  =  H(v)Vv  =  VL(v). 

(3)  If  the  field  is  both  solenoidal  and  lamellar, 

RVVL(v)  =  0,      whence      RVVv/(TVv)2  =  /(*), 
which  is  a  condition  on  the  character  of  the  curves.     Hence 


86  VECTOR  CALCULUS 

it  is  not  possible  to  have  a  solenoidal  and  lamellar  field 
with  purely  arbitrary  curves. 

(4)  If  the  field  is  solenoidal  and  Ta,  the  intensity, 
is  a  function  of  u  alone,  Y  =  p(u)/TVv,  and  therefore 
d  log  Y/dv  =  -  dTVv/TVvdv  =  -  RvVv/TVv2,  whence 

2RVW  =  d(TVv)2/dv, 

which  is  a  condition  on  the  curves.  An  example  is  the 
cross-section  of  a  field  of  magnetic  intensity  inside  an  in- 
finitely long  cylinder  of  revolution  which  carries  lengthwise 
a  steady  current  of  electricity  of  uniform  current  density. 

(5)  If  a  is  lamellar  and  Ta  is  a  function  of  v  only,  TVv 
=  g(v).  An  example  is  the  field  of  attraction  within  a 
homogeneous,  infinitely  long  cylinder  of  revolution.  The 
condition  is  a  restriction  on  the  possible  curves. 

(6)  If  the  field  is  lamellar  and  Ta  a  function  of  u  only, 
since  Y  is  a  function  of  v  only,  d  log  TVv/du  =  k(u),  or 
TVv  =  l(u)/m(v). 

This  restricts  the  curves. 

(7)  If  the  field  is  solenoidal  and  Ta  a  function  of  v  only, 
Ta  =  p(v)TW.  Therefore  d  log  Ta/dv  =  d  log  TVa/dv 
—  RS7Vv/(T\7v)2.  Hence  either  both  sides  are  constant 
or  else  both  expressible  in  terms  of  v.  If  the  field  is  not 
lamellar  also,  TVv  must  then  be  a  function  of  u  as  well  as 
of  v. 

(8)  If  the  field  is  lamellar  and  has  a  scalar  potential 
function,  that  is,  a  =  VP,  then  since  a  =  q(v)Vv,  we  must 
have  P  a  function  of  v  only,  and  a  =  P'Vfl.  From  this 
it  follows  that  diver  =  P\v)RVVv  +  P"(v)(TVv)2. 

(9)  If  the  field  is  uniform,  Ta  —  a,  Y  =  a/T\7v,  and  a 
is  lamellar  only  if  TVv  is  either  constant  or  a  function  of 
v  only,  while  a  is  solenoidal  only  if  we  have 

2RVW  =  d(TVv)2/dv. 


VECTORS   IN  A   PLANE  87 

(10)  Whatever  function  u  is,  the  u  lines  are  vector  lines 
for  the  vectors  £  =  f(u)UVv,  f  =  g(v)U\7v,  or 

T?  =  *(«,  r)tTVf. 

(11)  If  the  field  is  solenoidal,  TV  a  function  of  u  only, 
and  the  w  curves  are  the  lines  of  the  field,  then  the  curl 
takes  the  form  —  k  div  •  ka,  whence  it  has  the  form 

k[b(u)RVVu+  b'(u)(TVu)2], 

where  b  may  be  any  differentiate  function.     If  TV  is  also 
a  function  of  v,  the  form  of  the  curl  is 

k[b(u,  v)RVVu  +  db(u,  v)/du(TVu)2]. 

(12)  If  TV  is  a  function  of  u  only,  the  divergence  takes 
the  form 

diver  -  Ta[RWv/TVv  -  dTVv/dv]. 

(13)  If  TV  is  a  function  of  v  only 

curl  a  =  -  kTaTVu/TVv-dTVv/du. 

19.  Continuous  Media.  When  the  field  is  that  of  the 
velocity  of  a  continuous  medium,  we  have  two  cases  to 
take  into  account.  If  the  medium  is  incompressible  it  is 
called  a  liquid,  otherwise  a  gas.  Incompressibility  means 
that  the  density  at  a  point  remains  invariable,  and  if  this 
is  c,  then  from 

dc/dt=  dc/dt  +  RaVc,  =  dc/dt  +  RV(ca)  -  cRV<r 
we  see  that  the  first  two  terms  together  vanish,  giving  the 
equation  of  continuity,  since  they  give  the  rate  per  square 
centimeter  at  which  actual  material  (density  times  area, 
since  the  height  is  constant)  is  changing.     Hence  in  this  case 

dc/dt  =  —  cRV<t> 

This  gives  the  rate  of  change  of  the  density  at  a  point 
moving  with  the  fluid.  Hence  if  it  is  incompressible,  the 
velocity  is  solenoidal,  RV&  =  0. 


88  VECTOR  CALCULUS 

This  may  also  be  written  curl  (—  ka)  =  0,  hence  —  ka 
=  V?,  and  <j  —  kvQ,  which  shows  that  for  every  liquid 
there  is  a  function  Q  called  the  function  of  flow. 

When  curl  {  =  0,  we  have  seen  that  £  is  called  lamellar. 
It  may  also  be  called  irrotational,  since  the  curl  is  twice  the 
angular  rate  of  rotation  of  the  infinitesimal  parts  of  the 
medium,  about  axes  perpendicular  to  the  plane,  and  if 
curl  {  =  0  there  is  no  such  rotation.  Curl  is  analogous 
to  density,  being  a  density  of  rotation  when  the  vector 
field  is  a  velocity  field. 

The  circulation  of  the  field  is  the  integral  fRadp  along 
any  path  from  a  point  A  to  a  point  B.  This  is  the  same  as 
Xdx  +  Ydy,  and  is  exact  when 

dX/dy  =  dY/dx. 

But  this  gives  exactly  the  condition  that  the  curl  should 
vanish.  Hence  if  the  motion  is  irrotational  the  circulation 
from  one  point  to  another  is  independent  of  the  path.  In 
this  case  we  may  write  a  =  VP  where  P  is  called  the 
velocity  potential. 

When  a  is  irrotational,  the  lines  of  Q  have  as  orthogonals 
the  lines  of  P.  If  the  motion  is  rotational,  these  orthogonals 
are  not  the  lines  of  such  a  function  as  P.  If  the  motion  is 
irrotational,  we  have  for  a  liquid,  RwP  =  0,  and  P  must 
be  harmonic.  Hence  if  the  orthogonal  curves  of  the  Q 
curves  can  belong  to  a  harmonic  function  they  can  be  curves 
of  a  velocity  potential.  If  a  set  of  curves  belong  to  the 
harmonic  function  u,  then  RWu  =  0,  and  this  shows  that 
the  curl  of  —  JcVu  is  zero,  whence  Rdp(—  k\/u)  is  exact 
=  dv,  where  Vv  =  —  kVu.  From  this  we  have  Vm 
=  kvQ  for  the  condition  that  the  orthogonal  curves  belong 
to  a  harmonic  function.  This  however  gives  the  equation 
TS/u  =  TvQ.  We  may  assert  then  for  a  liquid  that  there 
is  always  a  function  of  flow,  and  the  curves  belonging  to 


VECTORS   IN   A   PLANE  89 

this  function  are  the  vector  lines  of  the  velocity,  the  in- 
tensity of  the  velocity  being  the  intensity  of  the  gradient  of 
the  function  of  flow.  If  the  orthogonal  curves  belong  to 
a  function  which  has  a  gradient  of  the  same  intensity,  both 
functions  are  harmonic,  the  function  of  the  orthogonal  set 
is  a  velocity  potential,  and  the  motion  is  irrotational. 

We  have  a  simple  means  of  discovering  the  sets  of  curves 
that  belong  to  harmonic  functions,  as  is  well  known  to 
students  of  the  theory  of  functions  of  a  complex  variable, 
since  the  real  and  the  imaginary  part  of  an  analytic  function 
of  a  complex  variable  are  harmonic  for  the  variable  co- 
ordinates of  the  variable.  That  is  to  say,  if  p  =  x  +  yk, 
and  £  =  /(p)  =  u  -\-  vk,  then  u,  v  are  harmonic  for  x,  y. 
The  condition  given  by  Cauchy  amounts  to  the  equation 
Vm  =  —  k\/v,  or  V£  =  0  where  £  is  a  complex  number. 
It  is  clear  from  this  that  the  field  of  £  is  both  solenoidal  and 
lamellar,  a  necessary  and  sufficient  condition  that  £  be  an 
analytic  function  of  a  complex  variable.  In  this  case  £  is 
called  a  monogenic  function  of  position  in  the  plane.  It  is 
clear  that  £  =  VH  where  H  is  a  harmonic  function. 

In  case  there  are  singularities  in  the  field  it  is  necessary 
to  determine  their  effect  on  the  integrals.  For  instance,  if 
we  have  a  field  a  and  select  a  path  in  it,  from  A  to  B,  or  a 
loop,  the  flux  of  a  through  the  path  is  the  integral  of  the 
projection  of  a  on  the  normal  of  the  path,  that  is,  if  the  path 
is  a  curve  given  by  dp,  so  that  the  projection  is  Ra(—  kdp), 
the  integral  of  this  is  the  flux  through, the  path.  It  is 
written 

2  =  SI  (-  Rakdp)  =  -  kfladp. 

In  the  case  of  a  liquid  the  condition  RV<r  =  0  shows  that 
the  expression  is  integrable  over  any  path  from  A  to  B, 
with  the  same  value,  unless  the  two  paths  enclose  a  singu- 
larity of  the  field.     In  the  case  of  a  node,  the  integral  around 

7 


90  VECTOR  CALCULUS 

a  loop  enclosing  the  node  is  called  the  strength  of  the  source 
or  sink  at  the  node.  We  may  imagine  a  constant  supply 
of  the  liquid  to  enter  the  plane  or  to  leave  it  at  the  node, 
and  be  moving  along  the  lines  of  the  field.  Such  a  system 
was  called  by  Clifford  a  squirt 

If  the  circulation  is  taken  around  a  singular  point  it 
will  usually  have  a  different  value  for  every  turn  around  the 
point,  giving  a  polydromic  function.  These  peculiarities 
must  be  studied  carefully  in  each  case. 

EXERCISES 

1.  From  £  =  Apn  we  find  in  polar  coordinates  that 

u  =  Arn  cos  nd,        v  =  Arn  sin  nd. 

These  functions  are  harmonic  and  their  curves  orthogonal.  Hence 
if  we  set  a  =  Vwora  =  V#,  we  shall  have  as  the  vector  lines  of  <r  the  v 
curves  or  the  u  curves.  What  are  the  curves  for  the  cases  n  =  —  3, 
—  2,  —  1,  1,  2,  3?     What  are  the  singularities? 

2.  Study  £  =  A  log  p,  and  £  =  A  log  (p  —  a)/(p  +  a). 

3.  Consider  the  function  given  implicitly  by  p  =  £  +  e*.  This 
represents  the  flow  of  a  liquid  into  or  out  of  a  narrow  channel,  in  the 
sense  that  it  gives  the  lines  of  flow  when  it  is  not  rotational. 

4.  Show  that  a  =  A/p  gives  a  radial  irrotational  flow,  while  a  =  Ak/p 
gives  a  circular  irrotational  flow.  What  is  true  of  a  =  Akpl  The 
last  is  Clifford's  Whirl. 

5.  Study  a  flow  from  a  source  at  a  given  point  of  constant  strength 
to  a  sink  at  another  point,  of  the  same  strength  as  the  source. 

6.  If  the  lines  are  concentric  circles,  and  the  angular  velocity  of  any 
particle  about  the  center  is  proportional  to  the  n-th  power  of  the  radius 
of  the  path  of  the  point,  show  that  the  curl  is  \  {n  +  2)  times  the  angular 
velocity. 

7.  A  point  in  a  gas  is  surrounded  by  a  small  loop.  Show  that  the 
average  tangential  velocity  on  the  loop  has  a  ratio  to  the  average 
normal  velocity  which  is  the  ratio  of  the  tensor  of  the  curl  to  the 
divergence. 

8.  What  is  the  velocity  when  there  is  a  source  at  a  fixed  origin,  and 
the  divergence  varies  inversely  as  the  w-th  power  of  the  distance  from 
the  origin.     [The  velocity  potential  is  A  log  r  —  B{n  —  2)~2r2-n.] 

9.  Consider  the  field  of  two  sources  of  equal  strength.  The  lines  are 
for  irrotational  motion,  cassinian  ovals,  where,  if  r,  r'  are  the  distances 


VECTORS   IN   A   PLANE  91 


from  the  two  sources  (foci)  and  rr'  —  h2,  Q  =  A  log  h  +  B,  the  velocity 
is  such  that  T<r  =  ATp/h2,  the  origin  being  half  way  between  the  foci; 
the  orthogonal  curves  are  given  by  u  =  iA[ir/2  —  (0  +  di)]  where  0, 
0i  are  the  angles  between  the  axis  and  the  radii  from  the  foci,  that  is 
they  are  equilateral  hyperbolas  through  the  foci.  The  circulation 
about  one  focus  is  ttA,  about  both  2irA. 

10.  If  the  lines  are  confocal  ellipses  given  by 

z2/m  +  i/VG*  -  c2)  =  1, 

then  Q  =  A  log  (  \V  +  V  (m  —  c2))  +  B.  If  p  is  the  perpendicular 
from  the  center  upon  the  tangent  of  the  ellipse  at  any  point,  then  the 
velocity  at  the  point  is  such  that  T<r  =  —  Ap/ y/  [/*(/*  —  c2)],  and 
the  direction  of  <r  is  the  unit  normal.  The  potential  function  is 
A  sin-1  B'  V  v\c.     V  v  is  the  semi-major  axis.    What  happens  at  the  foci? 

11.  If  the  stream  lines  are  the  hyperbolas  of  the  preceding,  then 
a  =  2 A  V  (*7(m  —  v))  times  the  unit  normal  of  the  hyperbola.  On  the 
line  p  =  yka  there  is  no  velocity,  at  the  foci  the  velocity  is  oo ,  half  way 
between  it  is  0.  The  lines  along  the  major  axis  outside  the  foci  act 
like  walls. 

12.  If  we  write  for  brevity  ux  for  T\7u,  and  vi  for  T\7v,  show  that 
we  have  whether  the  u  curves  are  orthogonal  to  the  v  curves  or  not, 

V  V  =  Ui2d2jdu2  +  Vi2d2ldv2  +  VVud/du  +  VVvd/dv 

+  2RVuVvd2/dudv. 

If  the  sets  of  curves  are  orthogonal  the  last  term  vanishes;  if  u  and  v 
are  harmonic  the  third  and  fourth  terms  drop  out;  if  both  cases  happen, 
only  the  first  two  terms  are  left. 

13.  In  case  of  polar  coordinates,  Vr  =  Up,  V0  =  r"2A;p  and 

VV  =  d2/dr2  +  r~ldldr  +  r_2d2/d02. 

14.  A  gas  moves  in  a  plane  in  lines  radiating  from  the  origin,  which 
is  a  source.  The  divergence  is  a  function  of  r  only,  the  distance  from 
the  center.     Find  the  velocity  and  the  density  at  any  point. 

a  =  pf{r),         flVo-  =  eir)  =  2/(r)  +  rf'{r), 
and 

f(r)  =  Ajr2  +  r~2fre{r)dr. 
To  determine  c, 

RV  log  co-  =  -  e(r)  =  f(r)Rp\/  log  c  =  rf(r)d  log  c/dr. 

15.  Show  that  in  the  steady  flow  of  a  gas  we  may  find  an  integrating 
factor  for  Rdpka  by  using  the  density,  [dc/dt  =  0  =  Rsjca  =  curl -fcco-, 
and  Rdpkca  is  exact.] 

16.  A  fluid  is  in  steady  motion,  the  lines  being  concentric  circles. 
The  curl  is  known  at  each  point  and  the  tensor  of  a  is  a  function  of  r 
only.     Find  the  velocity  and  the  divergence. 


92  VECTOR  CALCULUS 

17.  Rotational  motion,  that  is  a  field  which  is  not  lamellar,  is  also 
called  vortical  motion.  The  points  at  which  the  curl  does  not  vanish 
may  be  distributed  in  a  continuous  or  a  discontinuous  manner.  In 
fact  there  may  be  only  a  finite  number  of  them,  called  vortices.  We 
have  the  following: 

<r  =  k\7Q,        VVQ  =  T  curl  a  =  2«, 
Q  =  7r_1//«'  log  rdx'dy'  +  Q0, 

where  «'  denotes  co  at  the  variable  point  of  the  integration,  r  is  the 
variable  distance  from  the  point  at  which  the  velocity  is  wanted,  and  Q0 
is  any  solution  of  Laplace's  equation  which  satisfies  the  boundary 
conditions. 

If  the  mass  is  unlimited  and  is  stationary  at  infinity  we  have 

«  =  kfwfftt'ifi  -  P')/T(p  -  py-dx'dy'. 

A  single  vortex  filament  at  p  of  strength  I  would  give  the  velocity 

a  =U2T.(p-p')IT(p-p')\ 

If  we  multiply  the  velocity  at  each  point  p  at  which  there  is  a  vortex  by 
the  strength,  and  integrate  over  the  whole  field,  we  find  the  sum  is  zero. 
There  is  then  a  center  of  vortices  where  the  velocity  is  zero,  something 
like  a  center  of  gravity.     Instances  are 

(1)  A  single  vortex  of  strength  I.  The  vortex  point  will  remain  at 
rest,  and  points  distant  from  it  r  will  move  on  concentric  circles  with 
the  vortex  as  center,  and  velocity  l/2wr.  The  circulation  of  any  loop 
surrounding  the  vortex  is  of  course  the  strength. 

(2)  Two  vortices  of  strengths  k,  U.  They  will  rotate  about  the 
common  center  of  gravity  of  two  weighted  points  at  the  fixed  distance 
apart  a,  the  weights  being  the  two  strengths.  The  angular  velocity  of 
each  is 

27ra2 

The  stream  lines  of  the  field  are  given  by  fxhf%h  =  const.  When 
k  =  —  {,  the  center  is  at  infinity,  and  the  vortices  remain  a  fixed  dis- 
tance apart,  moving  parallel  to  the  perpendicular  bisector  of  this  segment 
joining  them.  Such  a  combination  is  called  a  vortex  pair.  The  stream 
lines  of  the  accompanying  velocity  are  coaxal  circles  referred  to  the 
moving  points  as  limit  points.  The  plane  of  symmetry  may  be  taken 
as  a  boundary  since  it  is  one  of  the  stream  lines,  giving  the  motion  of  a 
single  vortex  in  a  field  bounded  by  a  plane,  the  linear  velocity  of  the 
vortex  being  parallel  to  the  wall  and  \  of  the  velocity  of  the  liquid  along 
the  wall.  The  figure  suggests  the  method  of  images  which  can  indeed 
be  applied.  For  further  problems  of  the  same  character  works  on 
Hydrodynamics  should  be  consulted. 


VECTORS   IN  A   PLANE  93 

18.  Liquid  flows  over  an  infinite  plane  towards  a  circular  spot  where 
it  leaks  out  at  the  rate  of  2  cc.  per  second  for  each  cm.2  area  of  the  leaky 
portion.  The  liquid  has  a  uniform  depth  of  10  cm.  over  the  entire 
plane  field.  Find  formulas  for  the  velocity  of  the  liquid  inside  the 
region  of  the  leaky  spot,  and  the  region  outside,  and  show  that  there  is 
a  potential  in  both  regions. 

a  =  iVp  in  spot,  40/p  outside,  P  =  ^pp  in  spot,  40  log  Tp  —  20  log 
400  outside. 

Find  the  flux  through  a  plane  area  20  cm.  long  and  10  cm.  high,  whose 
middle  line  is  5  cm.  from  the  center  of  the  leaky  spot,  also  when  it  is 
30  cm.  from  the  leaky  spot.     Find  the  divergence  in  the  two  regions. 

Franklin,  Electric  Waves,  pp.  307-8. 

19.  Show  that  in  an  irrotational  motion  with  sources  and  sinks,  the 
lines  of  flow  are  the  orthogonal  curves  of  the  stream  lines  of  a  correspond- 
ing field  in  which  the  sources  and  the  sinks  are  replaced  by  vortices  of 
strengths  the  same  as  that  of  the  sources  and  sinks,  and  inversely. 
Stream  lines  and  levels  change  place  as  to  their  roles.  For  sources  and 
sinks  Q  =  1/2tt-ZZi0i,  P  =  1/2* -Z  log  rxh. 

20.  Vector  Potential.  In  the  expression  a  =  —  VkQwe 
express  (rasa  vector  derived  by  the  operation  of  V  upon 
—  JcQ,  the  latter  being  a  complex  number.  In  such  a  case 
we  may  extend  our  terminology  and  call  —  JcQ  the  vector 
potential  of  a.  A  vector  may  be  derived  from  more  than 
one  vector  potential.  In  order  that  there  be  a  vector 
potential  it  is  necessary  and  sufficient  that  the  divergence 
of  <t  vanish.  Hence  any  liquid  flow  can  have  a  vector 
potential,  which  is  indeed  the  current  function  multiplied 
by  —  k.     It  is  clear  that  Q  must  be  harmonic. 


CHAPTER  VI 

VECTORS   IN   SPACE 

1.  Biradials.  We  have  seen  that  in  a  plane  the  figure 
made  up  of  two  directed  segments  from  a  vertex  enables 
us  to  define  the  ratio  of  the  two  vectors  which  constitute 
the  sides  when  the  figure  is  in  some  definite  position.  This 
ratio  is  common  to  all  the  figures  produced  by  rotating  the 
figure  about  a  normal  of  the  plane  through  its  vertex,  and 
translating  it  anywhere  in  the  plane.  We  may  also  reduce 
the  sides  proportionately  and  still  have  the  same  ratio. 
The  ratio  is  a  complex  number  or,  as  we  will  say  in  general, 
a  hypernumber. 

If  now  we  consider  vectors  in  space  of  three  dimensions, 
we  may  define  in  precisely  the  same  manner  a  set  of  hyper- 
numbers  which  are  the  ratios  of  the  figures  we  can  produce 
in  an  analogous  manner.  Such  figures  will  be  called 
biradials.  To  each  biradial  there  will  correspond  a  hyper- 
number. Besides  the  translation  and  the  rotation  in  the 
plane  of  the  two  sides  of  the  biradial,  we  shall  also  permit 
the  figure  to  be  transferred  to  any  parallel  plane.  This 
amounts  to  saying  that  we  may  choose  a  fixed  origin,  and 
whatever  vectors  we  consider  in  space,  we  may  draw  from 
the  origin  two  vectors  parallel  and  equal  to  the  two  con- 
sidered, thus  forming  a  biradial  with  the  origin  as  vertex. 
Then  any  such  biradial  will  determine  a  single  hyper- 
number. Further  the  hypernumbers  which  belong  to  the 
biradials  which  can  be  produced  from  the  given  biradial 
by  rotating  it  in  its  plane  about  the  vertex  will  be  con- 
sidered as  equal. 

94 


VECTORS  IN  SPACE  95 

The  hypern umbers  thus  defined  are  extensions  of  those 
we  have  been  using  in  the  preceding  chapter,  the  new 
feature  being  the  different  hypernumbers  k  which  we  now 
need,  one  new  k  in  fact  for  each  different  plane  through  the 
given  vertex.  This  gives  us  then  a  double  infinity  of 
hypernumbers  of  the  complex  type,  r-cks  6,  where  the 
double  infinity  of  k's  constitute  the  new  elements. 

2.  Quaternions.  The  hypernumbers  we  have  thus  de- 
fined metrico-geometrically  involve  four  essential  param- 
eters in  whatever  way  they  are  expressed,  since  the 
biradials  involve  two  and  the  plane  in  which  they  lie  two 
more.  Hence  they  were  named  by  Hamilton  Quaternions. 
In  order  to  arrive  at  a  fuller  understanding  of  their  prop- 
erties and  relations,  we  will  study  the  geometric  properties 
of  biradials. 

In  the  first  place  if  we  consider  any  given  biradial,  there 
is  involved  in  its  quaternion,  just  as  for  the  complex  number 
in  the  preceding  chapter,  two  parts,  a  real  part  and  an 
imaginary  part,  and  we  can  write  the  quaternion  in  the 
form 

q  =  r  cos  6  +  r  sin  6 -a, 

where  a  corresponds  to  what  was  written  k  in  the  preceding 
chapter,  and  is  a  hypernumber  determined  solely  by  the 
plane  of  the  biradial.  On  account  of  this  we  may  properly 
represent  a  by  a  unit  normal  to  the  plane  of  the  biradial, 
so  taken  that  if  the  angle  of  the  biradial  is  considered  to  be 
positive,  the  direction  of  the  normal  is  such  that  a  right- 
handed  screw  motion  turning  the  initial  vector  of  the 
biradial  into  the  terminal  vector  in  direction  would  in- 
volve an  advance  along  the  normal  in  the  direction  in 
which  it  points.  It  is  to  be  understood  very  clearly  that 
the  unit  vector  a  and  the  hypernumber  a  are  distinct 
entities,    one   merely   representing   the    other.     The   real 


96  VECTOR  CALCULUS 

part  of  q  is  called,  according  to  Hamilton's  terminology,  the 
scalar  part  of  q,  and  written  Sq.  The  imaginary  part  is 
called,  on  account  of  the  representation  of  a  as  a  vector, 
the  vector  part  of  q  and  written  Vq.  The  unit  a  is  called  the 
unit  vector  of  q  and  written  UVq.  The  angle  of  q  is  0  and 
written  Zq.  The  number  r  which  is  the  ratio  of  the 
lengths  of  the  sides  of  the  biradial  is  called  the  tensor  of  q, 
and  written  Tq.  The  expression  cos  6  +  sin  6 -a  =  cas-d 
is  called  the  versor  of  q,  and  written  Z7^. 

Sq  is  a  quaternion  for  which  0  =  0°  or  180°,  Fg  is  a 
quaternion  for  which  0  =  90°  or  270°.  Tq  is  a  quaternion 
of  0°,  being  always  positive,  a  is  a  quaternion  of  0  =  90°, 
and  sometimes  called  a  right  versor. 

3.  Sum  of  Quaternions.  In  order  to  define  the  sum  of 
two  quaternions  we  define  the  sum  of  two  biradials  first. 
This  is  accomplished  by  rotating  the  two  biradials  in  their 
planes  until  their  initial  lines  coincide,  and  then  diminishing 
or  magnifying  the  sides  of  one  until  the  initial  vectors  are 
exactly  equal  and  coincide.  This  is  always  possible.  We 
then  define  as  the  sum  of  the  two  biradials,  the  biradial 
whose  initial  vector  is  the  common  vector  of  the  two,  and 
terminal  vector  is  the  vector  sum  of  the  two  terminal 
vectors.  The  sum  of  the  corresponding  quaternions  is 
then  the  quaternion  of  the  biradial  sum.  Since  vector 
addition  is  commutative,  the  addition  of  quaternions  is 
commutative. 

Passing  now  to  the  scalar  and  vector  parts  of  the  quater- 
nions, we  will  prove  that  they  can  be  added  separately,  the 
scalar  parts  like  any  numbers  and  the  vector  parts  like 
vectors. 

In  the  figure  let  the  biradial  of  q  be  OB/OA,  of  r  be 
OC/OA,  and  of  q  +  r  be  OD/OA.  Let  the  vector  part  of  q, 
Tq- sin  Zq-UVq  be  laid  off  as  a  vector  Vq  perpendicular 


VECTORS  IN  SPACE  97 

to  the  plane  of  the  biradial  of  q,  and  similarly  for  Vr. 
Then  we  are  to  show  that  V(q  +  r)  =  Vq  +  Vr  in  the 
representation  and  that  this  represents  the  vector  part  of 
q  +  r   according   to   the    definition.     It   is   evident   that 


OB  =  OB'  +  B'B,  the  first  vector  along  OA,  the  second 
perpendicular  to  OA.  Also  OC  =  OC"  +  C"&  +  C'C, 
the  first  part  along  OA,  the  second  parallel  to  B'B,  and  the 
third  perpendicular  to  the  plane  of  OAB.  The  sum 
OB  +  OC  =  OD,  where  OD  =  OB"  +  D"D'  +  D'Z),  and 
0Z>"  -  05'  +  00",  D"£'  =  B'B  +  C,,(7,,  D'D  =  C'C. 

Hence  the  biradial  of  the  sum  is  OD/OA,  where  the 
scalar  part  is  the  ratio  of  OD"  to  OA.  This  is  clearly  the 
sum  of  the  scalar  parts  of  q  and  r,  and 

S(q  +  f)  =  Sq+  Sr. 

The  vector  part  of  the  quaternion  for  OD/OA  is  the  ratio 
of  D"D  to  OA  in  magnitude,  and  the  unit  part  is  repre- 
sented by  a  unit  normal  perpendicular  to  OD"  and  D"D. 
But  D"D  =  B'B  +  C'C,  and  the  ratio  of  D"D  to  OA  equals 
the  sum  of  the  ratios  of  B'B  and  C'C  to  OA.  If  then  we 
draw,  in  a  plane  through  0  which  is  perpendicular  to  OA, 
the  vector  Vq  along  the  representative  unit  normal  of  the 
plane  OAB,  and  of  a  length  to  represent  the  numerical 
ratio  of  B'B  to  OA,  and  likewise  Vr  to  represent  the  ratio 
of  C'C  to  OA  laid  off  along  the  representative  unit  normal 


98  VECTOR   CALCULUS 

to  the  plane  OAC,  because  D"D  is  parallel  to  this  plane, 
as  well  as  B'B  and  C"C,  the  representative  unit  vector  of 
q+  r  will  lie  in  the  plane,  and  will  be  in  length  the  vector 
sum  of  Vq  and  Vr,  that  is  V(q  +  r)  as  shown. 

It  follows  at  once  since  the  addition  of  scalars  is  associa- 
tive, and  the  addition  of  vectors  is  associative,  and  the  two 
parts  of  a  quaternion  have  no  necessary  precedence,  that 
the  addition  of  quaternions  is  associative. 

4.  Product  of  Quaternions.  To  define  the  product  of 
quaternions  we  likewise  utilize  the  biradials.  In  this 
case  however  we  bring  the  initial  vector  of  the  multiplier 
to  coincide  with  the  terminal  line  of  the  multiplicand,  and 
define  the  product  biradial  as  the  biradial  whose  initial 
vector  is  the  initial  vector  of  the  multiplicand,  and  the 
terminal  vector  is  the  terminal  vector  of  the  multiplier. 
In  the  figure,  the  product  of  the  biradials  OB/OA,  and 


Fig.  13. 

OC/OB,  is,  writing  the  multiplier  first, 

OC/OB-  OB/OA  =  OC/OA. 

It  is  clear  that  the  tensor  of  the  product  is  the  product  of 
the  tensors,  so  that 

T-qr=  TqTr. 
It  follows  that 

U-qr  =  UqUr. 

It  is  evident  from  the  figure  that  the  angle  of  the  product 
will  be  the  face  angle  of  the  trihedral,  AOC,  or  on  a  unit 
sphere  would  be  represented  by  the  side  of  the  spherical 


VECTORS   IN   SPACE  99 

triangle  corresponding.  It  is  clear  too  that  the  reversal  of 
the  order  of  the  multiplication  will  change  the  plane  of 
the  product  biradial,  usually,  and  therefore  will  give  a 
quaternion  with  a  different  unit  vector,  though  all  the  other 
numbers  dependent  upon  the  product  will  remain  the  same. 
However  we  can  prove  that  multiplication  of  quaternions 
is  associative.  In  this  proof  we  may  leave  out  the  tensors 
and  handle  only  the  versors.  The  proof  is  due  to  Hamilton. 
To  represent  the  biradials,  since  the  vectors  are  all  taken 
as  unit  vectors,  we  draw  only  an  arc  on  the  unit  sphere, 
from  one  point  to  the  other,  of  the  two  ends  of  the  two  unit 
vectors  of  the  biradial.  Thus  we  represent  the  biradial 
of  q  by  CA,  or,  since  the  biradial  may  be  rotated  in  its 
plane  about  the  vertex,  equally  by  ED.  The  others  in- 
volved are  shown.  The  product  qr  is  represented  by  FD, 
from  the  definition,  or  equally  by  LM.  What  we  have 
to  prove  is  that  the  product  p  •  qr  is  the  same  as  the  product 
pq-r,  that  is,  we  must  prove  that  the  arcs  KG  and  LN  are 
on  the  same  great  circle  and  of  equal  length  and  direction. 


Fig.  14. 

Since  FE  =  KH,  ED  -  CA,  HG  =  CB,  LM  =  FD,  the 
points  L,  C,  G,  D  are  on  a  spherical  conic,  whose  cyclic 
planes  are  those  of  AB,  FE,  and  hence  KG  passes  through 
L,  and  with  LM  intercepts  on  AB  an  arc  equal  to  AB. 
That  is,  it  passes  through  N,  or  KG  and  LN  are  arcs  of  the 


100  VECTOR  CALCULUS 

same  great  circle,  and  they  are  equal,  for  G  and  L  are  points 
in  the  spherical  conic. 

5.  Trirectangular  Biradials.  A  particular  pair  of  bira- 
dials  which  lead  to  an  interesting  product  is  a  pair  of  which 
the  vectors  of  each  biradial  are  perpendicular  unit  vectors, 
and  the  initial  vector  of  one  is  the  terminal  of  the  other, 
for  in  such  case,  the  product  is  a  biradial  of  the  same  kind. 
In  fact  the  three  lines  of  the  three  biradials  form  a  tri- 
rectangular  trihedral.     If   the   quaternions   of   the   three 


o 

Fig.  15. 

are  i,  j,  k,  then  we  see  easily  that  the  quaternion  of  the 
biradial  OC/OB  is  represented  completely  by  the  unit  vector 
marked  i,  the  quaternion  of  OA/OC  by  j,  and  of  OB/OA  by 
k.    The  products  are  very  interesting,  for  we  have 

ij  =  k,        jk  =  i,        hi  =  j, 

and  if  we  place  the  equal  biradials  in  the  figure  we  also  have 

ji  =  —  k,         kj  =  —  i,         ik  =  —  j. 

Furthermore,  we  also  can  see  easily  that,  utilizing  the 
common  notation  of  powers, 

V-  =  -  1,       ?  -  -  1,       V  - ■■-  1. 

Since  it  is  evidently  possible  to  resolve  the  vector  part  of 
any  quaternion,  when  it  is  laid  off  on  the  unit  vector  of  its 
plane  as  a  length,  into  three  components  along  the  direc- 
tions of  i,  j,  k,  and  since  the  sum  of  the  vector  parts  of 


VECTORS   IN   SPACE 


10] 


quaternions  has  been  shown  to  be  the  vector  part  of  the 
sum,  it  follows  that  any  quaternion  can  be  resolved  into 
the  parts 

q  =  w  -\-  xi  -\-  yj  -\-  zk. 

These  hypernumbers  can  easily  be  made  the  base  of  the 
whole  system  of  quaternions,  and  it  is  one  of  the  many 
methods  of  deriving  them.  Hamilton  started  from  these. 
The  account  of  his  invention  is  contained  in  a  letter  to  a 
friend,  which  should  be  consulted.  (Philosophical  Maga- 
zine, 1844,  vol.  104,  ser.  3,  vol.  25,  p.  489.) 

6.  Product  of  Vectors.  It  becomes  evident  at  once  if  we 
consider  the  product  of  two  vector  parts  of  quaternions, 
or  two  quaternions  whose  scalar  parts  are  zero,  that  we 
may  consider  this  product,  a  quaternion,  as  the  product  of 
the  vector  lines  which  represent  the  vector  parts  of  the 
quaternion  factors.  From  this  point  of  view  we  ignore 
the  biradials  completely,  and  look  upon  every  geometric 
vector  as  the  representative  of  the  vector  part  of  a  set  of 
quaternions  with  different  scalars,  among  which  one  has 
zero  scalar.     From  the  biradial  definition  we  have 

VqVr=  S-VqVr+  V-VqVr 

equal  to  the  quaternion  whose  biradial  consists  of  two 
vectors  in  the  same  plane  as  the  vector  normals  of  the 


Fig.  16. 


102  VECTOR  CALCULUS 

biradials  of  Vq,  Vr  and  perpendicular  to  them  respectively. 
In  the  figure  the  biradial  of  Vr  is  OAB,  and  of  Vq  is  OBC, 
and  of  VqVr  is  OAC.  If  then  we  represent  the  vectors  by 
Greek  letters  whether  meant  to  be  considered  as  lines  or 
as  vector  quaternions,  a  =  Vq,  /3  =  Vr,  then  the  quaternion 
which  is  the  product  of  a(3  has  for  its  angle  the  angle  be- 
tween /3  and  a  +  180°,  and  for  its  normal  the  direction  OB. 
If  we  take  UVa(3  in  the  opposite  direction  to  OB,  and  of 
unit  length,  so  as  to  be  a  positive  normal  for  the  biradial 
a  /3  in  that  order,  then  we  shall  have,  letting  6  be  the  angle 
from  a  to  /3, 

a(3  =  TaTj3(-  cos  0  +  UVafi  sin  0). 

We  can  write  at  once  then  the  fundamental  formulae 

S-a&  =  -  TaTfi  cos  6,       V-a$  =  TaTp-sm  6-  UVaP. 

From  this  form  it  is  clear  also  that  any  quaternion  can 
be  expressed  as  the  product  of  two  vectors,  the  angle  of 
the  two  being  the  supplement  of  that  of  the  quaternion, 
the  product  of  their  lengths  being  the  tensor  of  the  quater- 
nion, and  their  plane  having  the  unit  vector  of  the  quater- 
nion as  positive  normal. 

If  now  we  consider  the  two  vectors  a  and  0  to  be  resolved 
in  the  forms 

a  =  ai-\-  bj  +  ck,        (3  —  li  +  mj  +  nk, 

where  i,  j,  k  have  the  significance  of  three  mutually  tri- 
rectangular  unit  vectors,  as  above,  then  since  Ta  Tfi  cos  6 
=  al-\-  bm-\-  en,  and  since  the  vector  Ta  T(3  sin  6  •  UVa(3 
is 

(bn  —  cm)i  +  (cl  —  an)j  +  (am  —  bl)k, 
we  have 

a/3  =  —  (al  +  bm  +  en)  +  (bn  —  cm)i 

+  (cl  —  ari)j  -\-  (am  —  bl)k. 


VECTORS   IN   SPACE  103 

But  if  we  multiply  out  the  two  expressions  for  a  and  0 
distributively,  the  nine  terms  reduce  to  precisely  these. 
Hence  we  have  shown  that  the  multiplication  of  vectors, 
and  therefore  of  quaternions  in  general,  is  distributive  when 
they  are  expressed  in  terms  of  these  trirectangular  systems. 
It  is  easy  to  see  however  that  this  leads  at  once  to  the 
general  distributivity  of  all  multiplications  of  sums. 

7.  Laws  of  Quaternions.  We  see  then  that  the  addition 
and  multiplication  of  quaternions  is  associative,  that 
addition  is  commutative,  and  that  multiplication  is  dis- 
tributive over  addition.  Multiplication  is  usually  not 
commutative.  We  have  yet  to  define  division,  but  if 
we  now  consider  a  biradial  as  not  being  geometric  but  as 
being  a  quaternion  quotient  of  two  vectors,  we  find  that 
P/a  differs  from  a(3  only  in  having  its  scalar  of  opposite 
sign,  and  its  tensor  is  T(3/Ta  instead  of  TaTfi. 

It  is  to  be  noticed  that  while  we  arrived  at  the  hyper- 
numbers  called  quaternions  by  the  use  of  biradials,  they 
could  have  been  found  some  other  way,  and  in  fact  were  so 
first  found  by  Hamilton,  whose  original  papers  should  be 
consulted.  Further  the  use  of  vectors  as  certain  kinds  of 
quaternions  is  exactly  analogous,  or  may  be  considered  to 
be  an  extension  of,  the  method  of  using  complex  numbers 
instead  of  vectors  in  a  plane.  In  the  plane  the  vectors 
are  the  product  of  some  unit  vector  chosen  for  all  the  plane, 
by  the  complex  number.  In  space  a  vector  is  the  product 
of  a  unit  vector  (which  would  have  to  be  drawn  in  the 
fourth  dimension  to  be  a  complete  extension  of  the  plane) 
by  the  hypernumber  we  call  a  vector.  However,  the  use  of 
the  unit  in  the  plane  was  seldom  required,  and  likewise  in 
space  we  need  never  refer  to  the  unit  1,  from  which  t^e 
vectors  of  space  are  derived.  On  the  other  hand,  just  as 
in  the  plane  all  complex  numbers  can  be  found  as  the  ratios 


104  VECTOR   CALCULUS 

of  vectors  in  the  plane  in  an  infinity  of  ways,  so  all  quater- 
nions can  be  found  as  the  ratios  of  vectors  in  space.  All 
vectors  are  thus  as  quaternions  the  ratios  of  perpendicular 
vectors  in  space.  And  multiplication  is  always  of  vectors  as 
quaternions  and  not  as  geometric  entities.  In  the  common 
vector  systems  other  than  Quaternions,  the  scalar  part  of 
the  quaternion  product,  usually  with  the  opposite  sign, 
and  the  vector  part  of  the  quaternion  product,  are  looked 
upon  as  products  formed  directly  from  geometric  con- 
siderations. In  such  case  the  vector  product  is  usually 
defined  to  be  a  vector  in  the  geometric  sense,  perpendicular 
to  the  two  given  vectors.  Therefore  it  is  a  function  of 
the  two  vectors  and  is  not  a  number  or  hypernumber  at 
all.  In  these  systems,  the  scalar  is  a  common  number,  and 
of  course  the  sum  of  a  number  and  a  geometric  vector 
is  an  impossibility.  It  seems  clear  that  the  only  defensible 
logical  ground  for  these  different  investigations  is  that  of 
the  hypernumber. 

It  is  to  be  noticed  too  that  Quaternions  is  peculiarly 
applicable  to  space  of  three  dimensions,  because  of  the 
duality  existing  between  planes  and  their  normals.  In  a 
space  of  four  dimensions,  for  instance,  a  plane,  that  is  a 
linear  extension  dependent  upon  two  parameters,  has  a 
similar  figure  of  two  dimensions  as  normal.  Hence,  corre- 
sponding to  a  biradial  we  should  not  have  a  vector.  To 
reach  the  extension  of  quaternions  it  would  be  necessary 
to  define  triradials,  and  the  hypernumbers  corresponding 
to  them.  Quaternions  however  can  be  applied  to  four 
dimensional  space  in  a  different  manner,  and  leads  to  a 
very  simple  geometric  algebra  for  four-dimensional  space. 
The  products  of  quaternions  however  are  in  that  case  not 
sufficient  to  express  all  the  necessary  geometrical  entities, 
and  recourse  must  be  had  to  other  functions  of  quaternions. 


VECTORS   IN  SPACE  105 

In  three-dimensional  space,  however,  all  the  necessary  ex- 
pressions that  arise  in  geometry  or  physics  are  easily 
found.  And  quaternions  has  the  great  advantage  over 
other  systems  that  it  is  associative,  and  that  division  is 
one  of  its  processes.  In  fact  it  is  the  most  complex  system 
of  numbers  in  which  we  always  have  from  PQ  =  0  the 
conclusion  P  =  0,  or  Q  =  0.* 

8.  Formulae.     It  is  clear  that  if  we  reverse  the  order  of 
the  product  ce/3  we  have 

0a  =  Soft  -  Vafi. 

This  is  called  the  conjugate  of  the  quaternion  a(3,  and 
written  K-a(3.     We  see  that 

SKq  =  Sq=  KSq,  VKq  =  -  Vq  =  KVq. 

Further,  since 

qr  =  SqSr  +  SqVr  +  SrVq  +  VqVr, 
we  have 

K-qr=  SqSr  -  SqVr  -  SrVq  +  VrVq  =  KrKq. 

From  this  important  formula  many  others  flow.  We  have 
at  once 

K-qi-  •  -qn  =  Kqn>  •  >Kqi. 
And  for  vectors 

Koli-  •  -0Ln  =  {—)nan-  •  •«!. 
Since 

Sq  =  i(q+Kq),  Vq=\{q-Kq), 

we  have  therefore 

S-OLl"  'Qt2n  =  i(<*l"  *  -«2n  +  «2n  '  *  'Oil), 

S-ai-  •  -C^n-l  =  i(tti*  •  'tt2n~l  —  «2n-l'  *  'Oil), 

V'CXi-  '  'OL2n  =  !(«!«  *  ,Q;2n   ~   «2n  '  "  '«t), 

F'Qfi-  •  ■a2n-l  =  %(<Xi'  '  -OL2n-\  +  «2n-l '  *  -«l). 

*  Consult  Dickson:  Linear  Algebras,  p.  11. 


106  VECTOR  CALCULUS 

In  particular 

2Sa$  =  aft  +  Pa,         2SaPy  =  afiy  -  y(3a, 
2Vap  =  a/3  -  fa         2Va(3y  =  a(3y  +  y(3a. 

It  should  be  noted  that  these  formulae  show  us  that  both 
the  scalar  and  the  vector  parts  of  the  product  can  them- 
selves always  be  reduced  to  combinations  of  products. 

This  is  simply  a  statement  again  of  the  fact  that  in 
quaternions  we  have'only'one  kind  of  multiplication,  which 
is  distributive  and  associative. 

We  see  from  the  expanded  form  above  for  S  •  qr  that 

S-qr  =  S-rq. 

Hence,  in  any  scalar  part  of  a  product,  the  factors  may  be 
permuted  cyclically.     For  instance, 

S-afi  =  S-(3a,        S-a(3y  =  S-Pya  =  S-yaQ, 
S-a(3y5  =  SPyfa 

From  the  form  of 

Sq=Uq+Kq),         Sq  =  SKq; 

hence  we  have 

Sa(3  =  S@a,         Safiy  =  -  Syfa         Sa(3y8  =  S8y(3a,  etc. 

From  the  form  of  VKq  =  —  Vq  we  see  that 

Vafi  =  -  V@a,         Vafiy  =  VyPa, 
Vapyh  m  -  Vdypa,         Vapyhe  =  VebyPa- 

We  do  not  have  a  simple  relation  between  V-qr  and 
V-rq,  but  we  have  the  fact  that  they  are  respectively  the 
sum  and  the  difference  of  two  vectors,  namely, 

If  a  —  SqVr  -+-  SrVq,  P  =  VVqVr,  then  ft  is  perpendicular 
to  a,  and 

Vqr  =  a  +  P,         Vrq  =  a  —  (3. 


q  =  w  + 

w?  +  , 

;V  +  ^ 

(Tq)2  =  w2  + 

x2  + 

2/2  +  *2, 

£g  = 

», 

(TTg)2  - 

z2  + 

f+z\ 

VECTORS   IN   SPACE  107 

It  is  obvious  that  TVqr  =  TVrq  and  that  /.qr  =  /rq 
-  tan-1  TVqr/Sqr.     The  planes  differ. 

The  product  of  g  and  i£g  is  the  square  of  the  tensor  of  q. 
We  indicate  the  unitary  part  of  q,  called  the  versor  of  q, 
by  Uq.     We  have  then  the  formulae 

Kq  =  w  —  ix  —  jy  —  kz, 
j j    =  w  +  ix  +  jy  +  kz 
q  Tq 

Vq  =  ix  +  jy  +  kz, 

TTVn  _ix  +  jy  +  fa 
^Kg rrg      ' 

(TVUq)2  m  (X*  +  f  +  *2)/(w2  +  a?  +  2/2  +  z2), 
cos-  Z   g  =  w/Tg  =  #•  £7g, 
sin-Z    g  =  TVq/Tq=  TVUq, 
Z-q=  tan"1  rFg/.Sg. 

The  product  of  two  quaternions  is 

qr  =  ww*  —  xx'  —  yy'  —  zz'  -f  i(wx'  -\-  w'x  +  yzf  —  y'z) 

+  j(wy'  +  w'y  +  zx'  —  z'x) 

+  k(wz'  +  w'a  +  xyf  —  x'y). 

From  the  formula  Tqr  =  TgTY  we  have  a  noted  identity 

(ief  +a*+  y2  +  z2)  <>'2+  x'2  +y'2  +  s'2) 
=  (wwf  —  ao'  —  2/2/'  —  zz')2  +  (wa;'  +  w'x  +  2/2'  —  S/'s)2 
+  (wy'  +  to'y  +  zx'  —  z'x)2  +  (W  +  w'z  +  #2/'  —  ^'2/)2- 

This  formula  expresses  the  sum  of  four  squares  as  the 
product  of  the  sums  of  four  squares.  It  was  first  given  by 
Euler.  The  problem  of  expressing  the  sum  of  three  squares 
as  the  product  of  sums  of  three  or  four  squares  and  the 
sum  of  eight  squares  as  the  product  of  sums  of  eight  squares 
has  also  been  considered. 


108 


VECTOR  CALCULUS 


9.  Rotations.     We  see  from  the  adjacent  figure  that  we 
have  for  the  product 

qrq-1 

a  quaternion  of  tensor  and  angle  the  same  as  that  of  r. 
But  the  plane  of  the  product  is  produced  by  rotating  the 
plane  of  r  about  the  axis  of  q  through  an  angle  double  the 
angle  of  q.  In  case  r  is  a  vector  /3  we  have  as  the  product 
a  vector  fif  which  is  to  be  found  by  rotating  conically  the 
vector  (3  about  the  axis  of  q  through  double  the  angle  of  q. 
It  is  obvious  that  operators*  of  the  type  qQq~l,  r()r-1, 
which  are  called  rotators,  follow  the  same  laws  of  multiplica- 
tion as  quaternions,  since  g(r()r_1)<7-1  =  qrQ[qr]~l.  A 
gaussian  operator  is  a  rotator  multiplied  by  a  numerical 
multiplier,  and  is  called  a  mutation.  The  sum  of  two 
mutations  is  not  a  mutation.  As  a  simple  case  of  rotator 
we  see  that  if  q  reduces  to  a  vector  a  we  have  as  the  result 
of  after1  =  /3'  the  vector  which  is  the  reflection  of  /3  in  a. 
The  reflection  of  /3  in  the  plane  normal  to  a  is  evidently 

—  a$orl. 

EXAMPLES 
(1)  Successive  reflection  in  two  plane  mirrors  is  equivalent 
*  QOq'1  represents  a  positive  orthogonal  substitution. 


VECTORS   IN  SPACE  109 

to  a  rotation  about  their  line  of  intersection  of  double  their 
angle. 

(2)  Successive  reflection  in  a  series  of  mirrors  all  per- 
pendicular to  a  common  plane,  2h  in  number,  making 
angles  in  succession  (exterior)  of  <pu,  (P23,  <&*•••  is  equivalent 
to  a  rotation  about  the  normal  to  the  given  plane  to  which 
all  are  orthogonal,  through  an  angle  6  =  2h  —  ir  —  2(<p12 
+  (pu  +  •••  +  <P2h-i,2h)  which  is  independent  of  the 
alternate  angles. 

(3)  Study  the  case  of  successive  reflections  in  mirrors  in 
space  at  any  angles. 

(4)  The  types  of  crystals  found  in  nature  and  possible 
under  the  laws  that  are  found  to  be  true  of  crystals,  are 
solids  such  that  every  face  may  be  produced  from  a  single 
given  face,  so  far  as  the  angles  are  concerned,  by  the 
following  op9rations : 

I,  the  reversal  of  a  vector,  in  quaternion 

form —  1 . 

A,  rotation  about  an  axis  a an()oTn. 

L4,  rotatory  inversion  about  a —  an()a~u. 

S,  reflection  in  a  plane  normal  to  /5 —  jSO/S-1  =  /?()/?. 

The  32  types  of  crystals  are  then  generated  by  the  succes- 
sive combinations  of  these  operations  as  follows: 

Triclinic  Ci    Asymmetric 1. 

d    Centre-symmetric 1,-1. 

Monoclinic        Cs    Equatorial 1,  0Q0. 

d    Digonal  polar 1,  a()a-1. 

C2h  Digonal  equatorial 1,  a()a;-1,  a()a. 

Orthorhombic   C2v  Didigonal  polar 1,  a()a~l,  0Q0,  Sap  =  0. 

D2   Digonal  holoaxial 1,  a()a-\  fiQfi'1,  Sap  =  0. 

Du  Didigonal  equatorial ....  1,    a()a-1,    POP'1,    «()«, 

SaP  =  0, 
A  =  al'20a-1'2. 
Tetragonal        d    Tetragonal  alternating  .  .1,  —  A. 

Du  Ditetragonal  alternating.  1,  —  A,  P{)P~X. 

d    Tetragonal  polar 1,  A. 


110  VECTOR  CALCULUS 

Ctk  Tetragonal  equatorial.  .  .1,  A,  aQa. 

C4*  Ditetragonal  polar 1,  A,  /3()/3. 

D4   Tetragonal  holoaxial ....  1,  A,  0Q0~K 

Dak  Dietragonal  equatorial  . .  1,  A,  aQa,  /3()/3_1. 
Rhombohedral  C8    Trigonal  polar l,B,  where  B  is  a2l30<*~il3' 

Czi  Hexagonal  alternating  .  .1,  B,  —  B. 

Ctv  Ditrigonal  polar 1,  B,  pQ0.    • 

D,   Trigonal  holoaxial 1,  B,  0Q0T+. 

Did  Dihexagonal  alternating .  1,    B,    j8()/8~l,     7O7,     7 

bisects  Z/3,  B0. 
Hexagonal         Czh  Trigonal  equatorial 1,5,  aQa. 

Dzh  Ditrigonal  equatorial  .  .  .1,  B,  aQa,  jS()/3_1. 

d    Hexagonal  polar 1,  C,  where  C  =  a1/3()«~1/3. 

dh  Hexagonal  equatorial  . . .  1,  C,  aQa. 

Civ  Dihexagonal  polar 1,  C,  /3()yS,  where  Sap  =  0, 

0  bisects  angle  of  7  and 
Cy,  Say  =  0. 

Di   Hexagonal  holoaxial  ....  1,  C,  /3()/S_1. 

Dan  Dihexagonal  equatorial.  .1,  C,  a()a,  pQ(3~l. 

Regular  T     Tesseral  polar ..1,     aQa'1,     PQP~X,     Safi 

=  Spy  =  Sya  =  0,     L 
where  L  =  (a  +  fj 

+  7)0(«+/3  +  7)-1. 

Th  Tesseral  central 1,  aQa~\  0Q/T1,   7O7"1, 

L,  aQa. 
Td   Ditesseral  polar 1,  aQa'1,  0Q0-\  7O7"1, 

L,  (a  +  fi)Q(a  +  /3). 
0     Tesseral  holoaxial 1,  aQa-\  0Q0~lt   yQy~l, 

L,  (a  +  p)Q(a  +  P)~K 
Oh   Ditesseral  central 1,  aQa~\  $00-*,  yQy'1, 

t,t{«  +  0)Q(a+0?t 

aQa. 

The  student  should  work  out  in  each  case  the  fuJl  set  of 
operators  and  locate  vectors  to  equivalent  points  in  the 
various  faces. 

Ref. — Hilton,  Mathematical  Crystallography,  Chap.  IV- 
VIII. 

(5)  Spherical  Astronomy.  We  have  the  following  nota- 
tion: 

X  is  a  unit  vector  along  the  polar  axis  of  the  earth, 
h  is  the  hour-angle  of  the  meridian, 


VECTORS  IN  SPACE  111 

L  =  cos  h/2  +  X  sin  h/2, 
i  =  unit  vector  to  zenith, 
j  =  unit  vector  to  south, 
k  =  unit  vector  to  east,  X  =  i  sin  I  —  j  cos  /,  where  I  is 

latitude, 
li  =  unit  vector  to  intersection  of  equator  and  meridian, 

\x  —  i  cos  I -\-  j  sin  I,  aSX/x  =  SkX  =  Sk/j,  =  0, 
d  =  declination  of  star, 

5  =  unit  vector  to  star  on  the  meridian  =  X  sin  d  +  jjl  cos  d, 
z  =  azimuth, 
A  =  altitude. 
At  the  hour-angle  h,  8  becomes  8'  =  L~l8L. 

The  vertical  plane  through  8f  cuts  the  horizon  in 
iVi8'  =  JSJ8'  +  kSk8',        tan  z  =  Sk8'/Sj8'. 
At  rising  or  setting  z  is  found  from  the  condition  Sid'  =  0. 
The  prime  vertical  circle  is  through  i  and  k.     The  6-hour 
circle  is  through  X  and  V\ji. 
a  —  right  ascension  angle, 
t  =  sidereal  time  in  degrees, 
h  =  t  +  a, 

Lt  =  cos  t/2  +  X  sin  t/2, 
La  =  cos  a/2  +  X  sin  a/2, 
e  =  pole  of  ecliptic, 

X  =  first  point  of  aries  =  vernal  equinox  =  Lrl^Lt} 
s  =  longitude, 
b  =  latitude, 
M  =  cos  s/2  +  e  sin  s/2. 

Problems.     Given  /,   d,  find  A  and  z  on  6-hour  circle. 
Sfx8'  =  0. 
/,  d,  find  h  and  z  on  horizon. 
/,  d,  find  A. 

I,  d,  A,  find  h  and  z,  8'  =  L~l8L  =  i 
cos  A  +  ?  cos  s  +  k  sin  2. 


112  VECTOR  CALCULUS 

/,  d,  h,  find  A  and  z. 
a  and  d,  find  s  and  b. 

(G)  The  laws  of  refraction  of  light  from  a  medium  of 
index  n  into  a  medium  of  index  n'  are  given  by  the  equation 

nVvct  —  n'Vva! 

where    v,  a,  a'  are  unit  vectors  along  the    normal,  the 
incident,  and  the  refracted  ray. 
The  student  should  show  that 

Investigate  two   successive  refractions,  particularly  back 
into  the  first  medium. 

(7)  It  is  easy  to  show  that  if  q  and  r  are  any  two  quater- 
nions, and  /3  =  V  •  VqVr,  we  may  write 

(8)  For  any  two  quaternions 

qiq'1  ±  r_1)  =  (r  =b  q)f\        and =  r(r  ±  q)~lq. 

-±  - 

9       r 

(9)  If  a,  b,  c  are  given  quaternions  we  can  find  a  quater- 
nion q  that  will  give  three  vectors  when  multiplied  by  a,  b, 
c  resp.     That  is,  we  can  find  q,  a,  ft  y  such  that 

aq  =  a,         bq  =  ft         eg  =  7.     (R.  Russell.) 

We  have  a  —  —  V •  Vc/aVa/b,  etc.,  or  multiples  of  these. 

(10)  In  a  letter  of  Tait  to  Cayley,  he  gives  the  following: 

(q+  r)()(g+  r)"1  =  (qlr)xrQf-i(q/r)-* 

=  qiq-iryQiq-^-vq-1  =  qh^Qq-^q-1, 
(Vq+  Vr)()(Vq+  Fr)"1  =  fa/rWJf^fo/r)-1/*, 


VECTORS   IN  SPACE  113 

where  tan  xA  =  a  sin  A/ (a  cos  A  +  1),  c  sin  2la  sin  ra/3 
+  cos  2la  cos  ra/3  =  2  (a  cos  o  +  &  cos  /S)  V  (6  sin  |8), 
2c  +  sin  2/a  cos  ra/3  =  2a  sin  a/ (6  sin  /3). 

Interpret  these  formulae. 

10.  Products  of  Several  Quaternions.  We  will  develop 
some  useful  formulae  from  the  preceding. 

If  we  multiply  a(3-(3a  we  have 

a2(32  -  S2a(3  -  V2a(3. 

Since  Sax  =  0,  if  x  is  a  scalar, 

&*/3t  =  SaVfry,         Sa(3y8  =  SaVfiyb,        etc. 
Since 

2Va(3  =  a(3  -  (3a,         2Sa(3  =  a(3  +  0ce, 
ffiaV(3y  =  af3y  —  ay  (3  —  (3ya  +  7/fa  =  2(7/3o  —  07/?) 

=  2(y(3a  +  7«/3  —  ay  {3  —  yap). 
For 

2<S/?7  •  a  =  /57a:  +  7/fo  =  2aSj3y  =  0:187  +  0:7/?, 
whence 

0:187  —  $70  =  Yj8o  —  07/?. 
Therefore 

VaV(3y  =  ySa(3  -  (3Say. 

Adding  to  each  side  ccSfiy,  we  have 

Va(3y  =  aS(3y  -  (3Sya  +  ySa(3. 
Since 

]S  =  crtaft         0  =  a^SaP  +  a~Wa$, 

which  resolves  (3  along  and  perpendicular  to  a, 

Sqrq-1  =  Sr  =  qSrq-1, 

Vqrq-1  =  h^q~l  -  Kq~lKrKq) 

=  iC^a-1  —  qKrq~l)  =  qVr-q~l. 

That  is,  if  we  rotate  the  field,  Sr  and  TTr  are  invariant. 


114  VECTOR   CALCULUS 

Hence  Vapy  =  VafiyaoT1  =  aV(3ya-oTl  and  Vafty, 
Vfiya.  are  in  a  plane  with  a  and  make  equal  angles  with  a. 
For  instance  if  a,  /?,  y,  Vafly,  Vfiya,  Vyafi  intersect  a 
sphere,  then  a,  /?,  y  bisect  the  sides  of  the  triangle  Vafiy, 
Vpya,  Vya(3,  a  being  opposite  to  Vya(3,  etc.  Evidently  if 
«i,  (X2-  •  -an  are  n  radii  of  a  sphere  forming  a  polygon,  then 
they  bisect  the  sides  of  the  polygon,  given  by  Vaia2-  •  -an, 
F«2«3-  •  '<xn,  Vets-  -  -anaia2,  •  • -Van(xi- - -an-i.  This  ex- 
plains the  geometrical  significance  of  these  vectors.  In 
fact  for  any  vector  a  and  quaternion  q,  the  vector  a  bisects 
the  angle  between  Vqa  and  Vaq,  that  is  to  say  we  construct 
Vqa  from  the  vector  Vaq  by  reflecting  it  in  a.  The  same 
is  true  for  any  product,  thus  (3yde  •  •  •  vol  is  different  from 
a(3y8e  •  •  •  v  only  in  the  fact  that  its  axis  is  the  reflection  in 
a  of  the  axis  of  the  latter. 

<M3 ' ' '  Qnqi  differs  from  qiq2  •  •  •  qn  only  in  the  fact  that 
its  axis  has  been  rotated  negatively  about  the  axis  of  q\ 
through  double  the  angle  of  qi.     Indeed 

?2?3-  •  -qnqi  =  q~Kqiq2-  ■  -qn)q\. 

If  we  apply  the  formula  for  expanding  VaVfiy  to 
V(Vafi)Vy8  =  —  V(Vy8)Va(3  we  arrive  at  a  most  im- 
portant identity: 

V-VapVy8  =  8Sa$y  -  ySa$8 

=  -  V-VydVafi  =  aS(3y8  -  /3Say8. 

From  this  equality  we  see  that  for  any  four  vectors 

8Sapy  =  aSfiyd  +  @Sya8  +  ySaj38. 

This  formula  enables  us  to  expand  any  vector  in  terms  of 
any  three  non-coplanar  vectors.     Again 
5Sapy  -  VpySad  =  V-aV(V(3y)8 

=  -  V-aV8V$y  =  Fa(3Sy8  -  VayS(38. 


VECTORS   IN  SPACE  115 

We  have  thus  another  important  formula 

SSofiy  =  Va(3Sy5  +  VfiySaB  +  VyaS08, 

enabling  us  to  expand  any  vector  in  terms  of  the  three 
normals  to  the  three  planes  determined  by  a  set  of  three 
vectors,  that  is,  in  terms  of  its  normal  projections.     Since 

aSPyS  =  VpySad  +  VytSefi  +  VbfiSay 

and 

(3Syda  =  Vay  S{38  +  VySSofi  +  VdaSPy, 
we  have 

VVapVyd  =  VabSPy  +  VPySad  -  VayS(38  -  VpbSay. 

From  this  we  have  at  once  an  expansion  for  Vafiyh,  namely 

Vctfyd  =  Va(3Sy8  -  VaySpb  +  VabSPy 

+  SapVyb  -  SayVpb  +  SabVpy. 
Also  easily 

Sapyd  =  SaPSyd  -  SaySpd  +  SabSpy. 
SVapVyb  =  SadSPy  -  SaySpb. 
V-ap-Sybe  =  yS-VapVbe  -  bS-VapVye  +  eS-  VapVyb 

y  b  e 

Say  Sab  Sae 

SPy  SPB  Spe 

In  the  figure  the  various  points  lie  on  a  sphere  of  radius  I. 
The  vectors  from  the  center  will  be  designated  by  the 
corresponding  Greek  letters.  The  points  X,  Y,  Z  are  the 
midpoints  of  the  sides  of  the  A  ABC.  From  the  figure  it 
is  evident  that 

H»  =  yli  =  (7/«1/2,         v/y  -  «h  m  (a/7)1*, 
Whence 

7  =  sar1,     «  -  nrr\     p  =  ^r1, 


116 


VECTOR  CALCULUS 


P 
Fig.  18. 
where 

v  =  it1!, 

and  the  axis  of  p  is  ±  a.  Also  p%p~l  =  ^iT1^7?-1*  so  that 
if  P  is  the  pole  of  the  great  circle  through  XY  then  the 
rotation  pQp~l  brings  £  to  the  same  position  as  the  rotation 
around  OP  through  twice  the  angle  of  tjJ-1.  Since  £  goes 
into  {'  by  a  rotation  about  OA  as  well  as  one  about  0Pf 
this  means  that  the  new  position  0Zr  is  the  reflection  of  OZ 
in  the  plane  of  OP  A.  The  angle  of  p  is  then  ZAL  or  ZAP 
according  as  the  axis  is  -\-  a  or  —  a.  The  angles  of  L  and 
M  are  right  angles,  and  if  we  draw  CN  perpendicular  to 
XY  then 


ANCY  =  ALAY,        ANCX  =  AMBX, 


and 


AL  =  BM  =  CN        and        APB        is  isosceles. 

Hence  the  equal  exterior  angles  at  A  and  B  are  ZAL 
=  ZBM  =  \{A  +  5  +  Q. 

Draw  PZ,  then  /ZiM  =  Zv^1  for  it  =JzJWM 
=  \ML  =  ZF  since  ilfZ  -  XN  and  iVF  =  YL.  The 
angle  between  the  planes  LAP  and  ZOP  is  thus  the  biradial 
7)%~l  and  also  £"  is  the  biradial  whose  angle  is  that  of  the 


VECTORS   IN   SPACE  117 

planes  OAZ,  ZOP,  so  that  ZOA  and  AOL  make  an  angle 
equal  to  z  p,  hence 

ZV  =  h(A  +  B+C). 
Further 

pa'1  -  nlyyfc'tla  =  («/t)1/2(t/«1/2(/3/«)1/2  -  p'. 

The  angle  of  p'  is  thus  %(A  +  5  +  C  -  tt)  =  2/2  where  S 
is  the  spherical  excess  of  AABC. 

Consider  the  quaternion  p  =  r)^1^  =  —  77^".  The  con- 
jugate of  p  is  Kp  =  ££77,  whose  axis  is  also  a  and  angle 

-  \{A  +  B  +  0).     Thus  the  quaternion  ffij  =  -  sin  2/2 

-  a:  cos  2/2. 

Shifting  the  notation  to  a  more  symmetric  form  we  have 
for  any  three  vectors 

aia2as  =  —  sin  2/2  —  TJVai(x2a.z  •  cos  2/2 

=  cos  \<j  —  k  sin  Jo-, 

where  2  is  the  spherical  excess  of  the  triangle  the  midpoints 
of  whose  sides  are  A\,  A2,  A%  and  a  is  the  sum  of  the  angles 
of  the  triangle.     Hence 

Saia2a3  =  cos  Jo",         Va ia2a3  =  ~*  UV<x\ol2ccz  sin  \a. 

It  is  to  be  noted  that  the  order  as  written  here  is  for  a 
positive  or  left-handed  cycle  from  A\  to  A2  and  A$.  Since 
2  is  the  solid  angle  of  the  triangle,  —  S-a\a2as  is  the  sine 
of  half  the  solid  angle  and  —  TVa\a2az  is  the  cosine  of  half 
the  solid  angle,  made  by  oi,  a2,  a3. 

If  now  we  have  several  points  as  the  middle  points  of  the 
sides  of  a  spherical  polygon,  say  aia2-  •  -an  and  the  vertex 
between  a\  and  an  is  taken  as  an  origin  for  spherical  arcs 
drawn  as  diagonals  to  the  vertices  of  the  polygon,  then  for 
the  various  successive  triangles  if  we  call  the  midpoints  of 
the  successive  diagonals 

J*lj   $2,    '  '  "fn-3 


118  VECTOR  CALCULUS 

we  have,  taking  the  axis  to  the  origin  which  we  will  call  k, 
and  which  is  the  common  axis  of  all  the  quaternions  made 
up  by  the  products  of  three  vectors 

The  sum  of  the  angles  of  the  polygon  is  the  sum  of  the 
angles  of  all  the  triangles  into  which  it  is  divided,  so  that 
if  this  sum  is  a  we  have  for  any  spherical  polygon 

«i«2-  •  *«n  =  (— )n_3[cos  cr/2  —  k  sin  a/2]. 

We  are  able  to  say  then  that  if  the  midpoints  of  the  sides 
of  a  spherical  polygon  are  ai,  a2,  •  •  -ant  then 

SoCi(X2'  '  '0in  =   db  COS  ff/2, 

where  a  is  the  sum  of  the  angles ;  the  vertices  of  the  polygon 
are  given  by 

Wolioli-  •  -an,         TJVcioOLz -  •  •  anai,  ••'•, 
UVan-  •  -ttn-l, 

each  being  the  vertex  whose  sides  contain  the  first  and  last 
vectors  in  the  product;  and  the  tensors  of  these  vectors  are 
each  equal  to  sin  <r/2. 

The  expression  —  Sa(3y  is  called  the  first  staudtian  of 
afiy,  the  second  staudtian  is 

-  SVapVPyVya/TVapTVPyTVya 

=  S2aj3y/TVaPTV(3yTVya, 

which  is  evidently  the  staudtian  of  the  polar  triangle. 

S-ai--an         ,iri        i 
mrz    — — — •  =  tan  f  solid  angle. 
1  V  •«!•  •  -an 

We  will  summarize  here  the  significance  of  the  expressions 
worked  out  thus  far,  and  in  particular  the  meaning  of  their 
vanishing. 


VECTORS   IN   SPACE  119 

Sa(3  is  the  product  of  TaTp  by  the  cosine  of  the  angle 
between  a  and  —  0.  It  vanishes  only  if  they  are  per- 
pendicular. 

Vafi  is  the  vector  at  right  angles  to  both  a  (3  whose  length  is 
TaTfi  multiplied  by  the  sine  of  their  angle.  It  vanishes 
only  if  they  are  parallel. 

Safiy  is  the  volume  of  the  parallelepiped  of  a  fi  y,  taken 
negatively.  It  vanishes  only  if  they  are  all  parallel  to 
one  plane. 

Vafiy,  Vafiyd,  •  •'•  these  vectors  are  the  edges  of  the  poly- 
hedral giving  the  circumscribed  polygon,  and  if  the  ex- 
pression vanishes,  we  have  by  separating  the  quaternion, 

Va0y8-  •  •  =  aS(3y8-  •  •  +  VaVPyS-'-  =  0. 

Hence  a  is  the  axis  of  (3yd-  •  •  and  Sfiyd-  •  •  equals  zero. 
By  changing  the  vectors  cyclically  we  have  n  vectors 
all  of  which  have  a  zero  tensor,  so  that  each  edge  is  the 
axis  of  the  quaternion  of  the  other  n  —  1  taken  cyclically. 
This  quaternion  in  each  case  has  a  vanishing  scalar. 
n  =  3,  a  j8  y  are  a  trirectangular  system. 
n  =  4,  a  (3  y  8  are  coplanar,  shown  by  the  four  vanish- 
ing scalars.     The  angle  a(3  =  angle  7#. 
n  =  5,  the  edge  Va(3y  is  parallel  to  V8e  and  cyclically 

similar  parallelisms  hold. 
We  have  in  all  these  cases  the  sum  of  the  angles  of  the 
circumscribing  polygon  a  multiple  of  2w  and  it 
satisfies  the  inequality  S(n  —  2)tt  is  greater  than 
a  which  is  greater  than  {n  —  2)x.  It  is  evident 
that  if  the  polygon  circumscribed  has  540°  the 
vectors  lie  in  one  plane.  ■ 
Safiyb  =  0.     If   e  =  Va(3y8,    then    VaQySe  =  0,    and    the 

preceding  case  is  at  hand  for  the  five  vectors. 
S-aia2-  •  -oLn  =  0,  the  sum  of  the  angles  of  the  polygon  is 
an  odd  multiple  of  x. 


120 


VECTOR  CALCULUS 


EXERCISES 

1.  S-VaPVpyVya  =  -  (Sapy)* 
V-VapVpyVya  =  VaP(y*SaP  -  SPySya)  +  ..... 

2.  S(a  +  P)iP  +  7)(7  +  «)  m  2Sa0y. 

3.  5-F(a  +  /3)(0  +  7)708  +  7)(7  +  a)V(y  +  «)(a  +  0) 

4.  5.F(Fa/3F/37)(F/37^7«)7(F7aFa/3)  =  -  (S-afiy)*. 

5.  S-5ef  -   -  16(5 -a^)4, 
where 

5  =  F(F[«  +  0[\fi  +  7]F[^  +  7][7  +  «]), 
<  =  y(7D9  +  7][7  +  a]V[y  +  a][a  +  fl), 
f  =  V(V[y  +  «][a  +  /S]7[a  +  0]\fi  +  7]). 

6.  S(xa  +  yP  +  27)(x'a  +  y'0  +  *'7)(x"a  +  y"0  +  *"7) 


4(5.afl7)1. 


7. 


x       \ 

X' 
X" 

S-aiPiyi  = 

- 

Saai     Sftai     Syai 
Sa0i     S00i     Syfii 
Say  1     Sfiyi     Syyi 

Saai     Sa&i 

S0ai   sm 

Syai     Syffi 
S8ai     S8P1 

s 
s 
s 

,8 

ayi     Sadi 
Pyi     SP81 
771     Sy8i 
571      S881 

■■  0 

S  •  a/37. 


for  any  eight  vectors.     If  the  element  Saai  is  changed  to  Szai  the  value 
is  -  S-0y8'S'Piyi8i-S-ai(e  —  a). 

9.  S-Va0yV0yaVyaP  =  ISaPSPySyaSaPy. 

10.  From  S2P/a  -  V2Pfp  =  1  we  find 


T(Sp/a  +  Vp/P) 
where 


1  =  T{\cl+p  +  \pa~i 


-  irv  +  yr1)  =  T(a'P  +  p/80 

a'  =  §(«T*  -  r>),         pV  =  i(a"»  +  p*-«). 

11.  If  TP  =  Ta  =  Tp  =  1  and  S-afip  =  0, 

S-U(p-a)U(p  -P)  =  ±iV[2(l  -Sap)]. 

12.  If  a,  P,  7  and  ah  Pi,  71  are  two  sets  of  trirectangular  unit  vectors 
such  that  if  a  =  Py,  ax  =  Piy,  then  we  may  find  angles  called  Eulerian 
angles  such  that 

a2  =  a  COS  yp  +  P  sin  \J/,  P2   =    —  a  sin  i£  +  P  COS  ^, 

73  =  7  cos  6  +  <*2  sin  0,         a3  =  —  7  sin  0  -f  «2  cos  0, 

71  =  73,         «i  =  «3  cos  ^  +  /?2  sin  <p, 

Pi  =  —  a3  sin  v>  +  /32  cos  <p. 


VECTORS   IN  SPACE  121 

13.  If  q  =  ai«2  •  •  •  otn  then  if  we  reflect  an  arbitrary  vector  in 
succession  in  a„,  an-i,  •  •  •  0:20:1  when  Sq  =  0  the  final  position  will  be  a 
simple  reflection  of  p  in  a  fixed  vector,  and  if  Vq  =  0  the  final  position 
will  be  on  the  line  of  p  itself.  Similar  statements  hold  if  the  reflections 
are  in  planes  that  are  normal  respectively  to  an,  •  •  •  «i. 

11.  Functions.  We  notice  some  expressions  now  of  the 
nature  of  functions  of  a  quaternion.  We  have  the  follow- 
ing identity  which  is  useful : 

(a/3)n  +  {$a)n  =  (ol$  +  $a)l(<xP)n~l]  ~  a^a[(a^n~2 

=  2SaP[(a(3)n~1  +  (/to)71-1]  -  a2^[(a^n~2 

+  08*)*-*]. 

Whence  2nSna(3  =  (a/3  +  Mn  =  [(«/3)n  +  (fax)"] 

+  lt/  nl  ni  K«/5)"-2  +  (/3a)""2]a2/32 
\\{n  —  1)1 

+ 2l(nwl2)1  [(«»n_4  +  w-v/34  +  •  •  • 

\\{n  —  1)1 

This  implies  the  familiar  formula  for  the  expansion  of  cosn  0 
in  terms  of  cos  nd,  cos  (n  —  2)0,  and  we  can  write  as  the 
reverse  formula 

S(a(3)n  -  (-)w/2[an/3n  -  n2S2a(3-an-2l3n-2l2\ 

+  n2(n2  -  22)SV-an"4/3"-4/4!  -  •  •  •]  n  even 
(- )  (n~l)  '2[nSa(3  •  an~ler-lll ! 

-  n(n2  -  l2)53a/5-o:n-3/3n-3/3!  +  •  •  •]  n  odd. 
Likewise 


TV2na$=  (-l)n/22n-1[S(al32n 

(2n)! 


l!(2n-  1) 


S(aP2n~2a2p2  +  ■••] 


122  VECTOR   CALCULUS 

7»p»-ia/3==  (_l)«/22«-2[7T(a/3)2n-1 

-    (2n  -  ^l  TV(aB)2n~3  +  • .  .1 
l!(2n-2)1 1VKfxp)       x        J 

TV(ap)n/TVap  =  (-)n/2[n5aiS-Q:n-2/?n-2/l! 

-  n(n2  -  22)iS3a/3«n-4/Sn~4/3!  +  •  •  -J  n  even 
(_1)<*-*^1  -  (n2  -  l^SPap-cT+p^fil  +  •  •  •]  n  odd. 

Since  jS/a  is  a  quaternion  whose  powers  have  the  same 
axis  we  have  (1  —  0/a)-1  =  1  +  fi/a  +  03/a:)2  +  •  •  •  when 
Tfi  <  Ta,  and  taking  the  scalar  gives  the  well-known 
formula 


Likewise 


S-^~=  1  +  S/5/a  +  S(/3/a)2  + 
a  —  p 


TV-^—=  TVp/a  +  TV(p/a)2  + 
a  —  p 


If  we  define  the  logarithm  as  in  theory  of  functions  of  a 
complex  variable  we  have 

log  (1  -  fi/a)  =  log  7(1  -  fi/a)  +  log  17(1  -  fi/a) 

=  -  fa  -  Itf/a)*  -  HP/a)* . 

Therefore 

log  f(l  -  fi/a)  -  -  Sfi/a  -  §S(/?/c*)2 

Z  °LZ_1  =  TV  log  (1  -  fi/a)  =  TVp/a  +  ^TV(p/a)2- 
a 

Again 

T{a  -  p)~l  =  Ta'1  -  f(l  -  P/a)-1  -  fo^l  + 

Pi(-  SUp/a) TP/a  +  P2(-  SUp/a) T2P/a  +  .••], 

where  Pi  P2  are  the  Legendrian  polynomials. 

Evidently  for  coaxial  quaternions  we  have  the  whole 
theory  of  functions  of  a  complex  variable  applicable. 


VECTORS   IN  SPACE  123 

12.  Solution  of  Some  Simple  Equations. 

(1).  If  ap  =  a  then  p  =  oTla. 

(2) .  If  Sap  =  a  then  we  set  Vap  =  f  where  £*  is  any  vector 
perpendicular  to  a,  and  adding,  p  =  aa_1  +  a~l$. 

(3).  If  Fap  =  jS  then  *Sap  =  a:  where  #  is  any  scalar,  and 
adding  we  have  p  =  a~l(3  +  aaaf"1. 

(4).  If  Vapfi  =  y  then  SaVapQ  =  &x2p/3  =  <*2£p/3  =  Say 
and  SpVap(3  =  /32£ap  =  S/fy.     Now 

Fap/5  =  aS/3p  -  pSafi  +  (3Sap 

and  substituting  we  have 

p  =  [o;-1^7  +  /T1^  -  y]/8afi. 

The  solution  fails  if  Sa(3  =  0.     In  this  case  the  solution  is 

p  =  _  a-'S^y  -  p^Sa-iy  +  xVofi, 

x  any  scalar. 

(5).  If  Yapp  =  7  then  Sa(3pSafi  =  &*07  and  Soft) 
=  Sa(3y/Sa(3.     Adding  to  Va(3p,  we  have 

afip  =  7  +  Sa(3y/SaP  and  p  =  0^or*7  +  '(hcT*8cfiyl8c&. 

(6).  If  &xp  =  a,  £/3p  =  b,  then  a^p  =  zFa/3  +  V(al3 
-  ba)Va(3. 

(7).  If  Sap  =  a,  S(3p  =  b,  Syp  =  c,  then 

pSafiy  =  aV(3y  +  bVya  +  cFa/5. 

(8).  If  gag-1  =  |3  then  g  =  (x/3  +  y)/(a  +  /3)  where  x  and 
?/  are  any  scalars.     Or  we  may  write 

q  =  u  +  0(a  +  |8)  +  wFa?/3      where      u  =  —  w#a(a:  +  /3). 

(9).  If  gag"1  =  y,  q^q'1  =  8,  then 

V(y  -  a)(8  -  ft! 


.. 


1  + 


S(T  +  «)(«-  ft 


124  VECTOR  CALCULUS 

(10).  If  qaq-1  =  f,  qpq~l  =  *,  qyq~l  =  f,  then 

S-flft  -  «)  -  0,  S-q(V  -  ft  -  0,  flf.gtf  -  7)  =  0, 

hence  Fg  is  coplanar  with  the  parentheses,  and  we  have 

x(i  -  a)  +  2/(77  -  ft  +  H(f  -  7)  =  0 
where 

»:*:*->-  2S7(r?  -  ft  :  2Sy(i  -  a)  :  S(£  +  a)(i,  -  ft. 

The  six  vectors  are  not  independent.     Vq  is  easily  found 
and  thence  Sq  from 

qa  =  £q. 

(11).  If  (p  -  a)"1  +  (p  -  ft"1  -  (P  ~  7)"1  ~  (P  ~  5)-1 
=  0,  then  if  we  let 

ifi'  ~  aT1  =  1  *  (TO  -  5)"1  -  5]  -  [(a  -  6)"1  -  5]) 

=  (p  —  8)(p  —  a)_1(«  —  5),  etc., 

where  p',  a',  0',  7'  are  the  vectors  from  D,  the  extremity  of 
5,  to  the  inverses  with  respect  to  D,  of  the  extremities  of 
p,  a,  ft  7,  then 

(p'  -  a')"1  +  (p'  -  ft)"1  -  (p'  -  7T1  =  0. 

Prove  that 


1  -  ft  _  y  -  ft  _  P'  -  y  _  r  y  -  /ni/2 


p 


whence  p'  and  p.  (R.  Russell.) 

(12).  If  (q  -  a)"1  +  (q  -  6)"1  -  (q  -  c)"1  -  (q  -  d)~' 
=  0,  we  set 

(q  -  d)(q'  -  d)=  (a-  d){a'  -  d)  =  (b  -  d)(b'  -  d) 

=  (c  -  d){c'  -  d)  -  1, 


VECTORS    IN   SPACE  125 

thence 

(q  -  d)->  -(q-  a)'1  =  (4  -  d)-\a  -  d)(q  -  a)'1 
(q  -  d)~i  [(a~d)/(q-a)+(b-  d)l(q-  b)-  (c  -  d)/(q-  e)] 
-  (?'  -  a')'1  +  (?'  -  &T1  ~  W  ~  cT1 
and  we  have  q'  from 

(V  -  cW  -  C)  =  (g'  -  6')/(g'  -  «0 

=  (q'  -  c')l(a'  -  c')  -  [(V  -  c')Ka'  -  c')]K 

(R.  Russell.) 

13.  Characteristic  Equation.  If  we  write  q  =  Sq  +  Vq 
and  square  both  sides  we  have  q2  =  S2q  +  (Vq)2  +  2Sq-Vq 
whence 

g2  -  2qSq  +  S2q  -  V2q  =  0. 

This  equation  is  called  the  characteristic  equation  of  q. 
The  coefficients 

2Sq         and         S2q  -  V2q  =  T2q 

are  the  invariants  of  q;  they  are  the  same,  that  is  to  say, 
if  q  is  subjected  to  the  rotation  r()r-1.  They  are  also  the 
same  if  Kq  is  substituted  for  q.  Hence  they  will  not  define 
q  but  only  any  one  of  a  class  of  quaternions  which  may  be 
derived  from  each  other  by  the  group  of  all  rotations  of  the 
form  rQr~l  or  by  taking  the  conjugate. 
The  equation  has  two  roots  in  general, 

Sq  +  Tqyl  -  1         and         Sq  -  Tq^  -  1. 

Since  these  involve  the  V  —  1  it  leads  us  to  the  algebra  of 
biquaternions  which  we  do  not  enter  here,  but  a  few  re- 
marks will  be  necessary  to  place  the  subject  properly. 

Since  the  invariants  do  not  determine  q  we  observe  that 
we  must  also  have  UVq  in  order  to  have  the  other  two 
parameters  involved. 


126  VECTOR  CALCULUS 

If  we  look  upon  UVq  as  known  then  we  may  write  the 
roots  of  the  characteristic  equation  in  the  number  field  of 
quaternions  as  Sq  +  TVqUVq  and  Sq  —  TVqUVq  or 

q        and         Kq. 

If  we  set  q  -f-  r  for  q  and  expand,  afterwards  drop  all  the 
terms  that  arise  from  the  identical  equations  of  q  and  r 
separately,  we  have  left  the  characteristic  equation  of  two 
quaternions,  which  will  reduce  to  the  first  form  when  they 
are  made  to  be  equal.     This  equation  is 

qr+rq-2Sq-r-  2Sr-Vq  +  2SqSr  -  2SVqVr  =  0. 

We  might  indeed  start  with  this  equation  and  develop  the 
whole  algebra  from  it. 
We  may  write  it 

qr-\-  rq-  2qSr  -  2rSq  +  4Sq-Sr  +  S-qr  +  S-rq  =  0 

which  involves  only  the  scalars  of  q,  r,  qr,  and  rq. 

14.  Biquaternions.  We  should  notice  that  if  the  param- 
eters involved  in  q  can  be  imaginary  or  complex  then 
division  is  no  longer  unique  in  certain  cases.     Thus  if 

Q2=q2 
we  have  as  possible  solutions 

Q  =  ±  q        and  also  Q  =  ±  V  (-  l)UVq-q. 

If  q2  =  0  and  Vq  =  0  then  TVq  =  0  and  we  have 
Vq  =  x(i  +  j  V  —  1)  where  X  is  any  scalar  and  i,  j  are  any 
two  perpendicular  unit  vectors. 


CHAPTER  VII 
APPLICATIONS 

1.  The  Scalar  of  Two  Vectors 
1.  Notations.  The  scalar  of  the  product  of  two  vectors 
is  defined  independently  by  writers  on  vector  algebra,  as 
a  product.  In  such  cases  the  definition  is  usually  given  for 
the  negative  of  the  scalar  since  this  is  generally  essentially 
positive.  A  table  of  current  notations  is  given.  If  a  and  (3 
define  two  fields,  we  shall  call  S*cfi  the  virial  of  the  two 
fields. 

S-a(3  =  —  a  X  /3     Grassman,  Resal,  Somoff,  Peano,  Bura- 
li-Forti,  Marcolongo,  Timerding. 

—  Cfft         Gibbs,  Wilson,  Jaumann,  Jung,  Fischer. 

—  a/3  Heaviside,  Silberstein,  Foppl,  Ferraris, 

Heun,  Bucherer. 

—  (aft)        Bucherer,    Gans,    Lorentz,    Abraham, 

Henrici. 

—  a|/3         Grassman,  Jahnke,  Fehr,  Hyde. 
Cos  a/3         Macfarlane. 

[a/3]  Caspary. 

For  most  of  these  authors,  the  scalar  of  two  vectors, 
though  called  a  product,  is  really  a  function  of  the  two 
vectors  which  satisfies  certain  formal  laws.  While  it  is 
evident  that  any  one  may  arbitrarily  choose  to  call  any 
function  of  one  or  more  vectors  their  product,  it  does  not 
seem  desirable  to  do  so.  For  Gibbs,  however,  the  scalar 
is  defined  to  be  a  function  of  the  dyad  of  the  two  vectors, 
which  dyad  is  a  real  product.  The  dyad  or  dyadic  of 
Gibbs,  as  well  as  the  vectors  of  most  writers  on  vector 
analysis,  are  not  considered  to  be  numbers  or  hypernumbers. 

127 


128  VECTOR  CALCULUS 

They  are  looked  upon  as  geometric  or  physical  entities, 
from  which  by  various  modes  of  "combination"  or  de- 
termination other  geometric  entities  are  found,  called 
products.  The  essence  of  the  Hamiltonian  point  of  view, 
however,  is  the  definition  by  means  of  geometric  entities  of 
a  system  of  hypernumbers  subject  to  one  mode  of  multiplica- 
tion, which  gives  hypernumbers  as  products.  Functions 
of  these  products  are  considered  when  useful,  but  are  called 
functions. 

2.  Planes  and  Spheres.  It  is  evident  that  the  condition 
for  orthogonality  will  yield  several  useful  equations,  and 
of  these  we  will  consider  a  few. 

The  plane  through  a  point  A,  whose  vector  is  a,  per- 
pendicular to  a  line  whose  direction  is  8  has  for  its  equation, 
since  p  —  a  is  any  vector  in  the  plane, 

S-d(p-a)  =  0. 

If  we  set  p  =  8Sa/d  we  have  the  equation  satisfied  and  as 
this  vector  is  parallel  to  5  it  is  the  perpendicular  from  the 
origin  to  the  plane.  The  perpendicular  from  a  point  B 
is  b~lS{a  -  0)5. 

If  a  sphere  has  center  D  and  radius  T(3  where  /?  and  —  (3 
are  the  vectors  from  the  center  to  the  extremities  of  a 
diameter,  then  the  equation  of  the  sphere  is  given  by  the 
equation 

S(p  -  3  +  fi)(p  -  d  -  P)  =  0,  orp2  -  2S8P  +  52  -  /32  =  0. 

The  plane  through  the  intersection  of  the  two  spheres 

p2  -  2£5ip  +  ci  =  0  =  p2  -  2S82P  +  c2 

is  2S(5i  —  52)p  =  ci  —  c2. 

The  form  of  this  equation  shows  that  it  represents  a  plane 


APPLICATIONS  129 

perpendicular  to  the  center  line  of  the  spheres.    The  point 
where  it  crosses  this  line  is 

X18]  +  x282 

P  = i » 

Xi  +  x2 

whence  solving,  we  find 

p  =  v(h  +  82)-\V8,82  +  i(cj  -  <*)>. 

3.  Virial.  If  (3  is  the  representative  of  a  force  in  direction 
and  magnitude  then  its  projection  on  the  direction  a  is 
a~1Sa^f  and  perpendicular  to  this  direction  crWafi.  If  a 
is  in  the  line  of  action  of  the  force,  the  projection  is  fit  If  a 
is  a  direction  not  in  the  line  of  action  then  the  projection 
gives  the  component  of  the  force  in  the  direction  a.  If  a 
is  the  vector  to  the  point  of  application  of  the  force  then 
Sa(3  is  the  virial  of  the  force  with  respect  to  a,  a  term  intro- 
duced by  Clausius.  It  is  the  work  that  would  be  done  by 
the  force  in  moving  the  point  of  application  through  the 
vector  distance  a.  If  a  fe  an  infinitesimal  distance  say, 
8a,  then  —  S8a(3  is  the  virtual  work  of  a  small  virtual  dis- 
placement. The  total  virtual  work  would  be  8V  = 
—  2S8an(3n  for  all  the  forces. 

4.  Circulation.  In  case  a  particle  is  in  a  vector  field 
(of  force,  or  velocity,  or  otherwise)  and  it  is  subjected  to 
successive  displacements  8p  along  an  assigned  path  from 
A  to  B,  we  may  form  the  negative  scalar  of  the  vector 
intensity  of  the  field  and  the  displacement.  If  the  vector 
intensity  varies  from  point  to  point  the  displacements 
must  be  infinitesimal.  The  sum  of  these  products,  if  there 
is  a  finite  number,  or  the  definite  integral  which  is  the  limit 
of  the  sum  in  the  infinitesimal  case,  is  of  great  importance. 
If  a  point  is  moving  with  a  velocity  a  [cm./sec]  in  a  field  of 
force  of  /3  dynes,  the  activity  of  the  field  on  the  point  is 


130  VECTOR  CALCULUS 

—  S-(3<t  [ergs/sec.].  The  field  may  move  and  the  point 
remain  stationary,  in  which  case  the  activity  is  S-(3a.  The 
activity  is  also  called  the  effect,  and  the  power.  If  <r  is  the 
vector  function  of  p  which  gives  the  field  at  the  point  P  we 
have  for  the  sum 

-  2Sa8p        or         -  //  Sa8p. 

This  integral  or  sum  is  called  the  circulation  of  the  path  for 
the  field  a. 

5.  Volts,  Gilberts.  For  a  force  field  the  circulation  is  the 
work  done  in  passing  from  A  to  B.  If  the  field  is  an  electric 
field  E,  the  circulation  is  the  difference  in  voltage  between 
A  and  B.  If  the  field  is  a  magnetic  field  H,  then  the  circula- 
tion is  the  difference  in  gilbertage  from  A  to  B.  It  is 
measured  in  gilberts,  the  unit  of  magnetic  field  being  a 
gilbert  per  centimeter.  There  is  no  name  yet  approved  for 
the  unit  of  the  electrostatic  field,  and  we  must  call  it  volt 
per  centimeter.  The  unit  of  force  is  the  dyne  and  of  work 
the  erg. 

6.  Gausses  and  Lines.  In  case  the  field  is  a  field  of  flux 
a,  and  the  vector  TJv  is  the  outward  normal  of  a  surface 
through  which  the  flux  passes,  then 

-  SaUv 

is  the  intensity  of  flux  normal  to  or  through  the  surface 
per  square  centimeter.  The  unit  of  magnetostatic  flux  B 
is  called  a  gauss;  the  unit  of  electrostatic  flux  D  is  called  a 
line.  The  total  flux  through  a  finite  surface  is  the  areal 
integral 

—  fSaUvdA,        written  also         —  fSadv. 

The  flux-integral  is  called  the  transport  or  the  discharge. 
Thus  if  D  is  the  electric  induction  or  displacement,  the 


APPLICATIONS  131 

discharge  through  a  surface  A  is  —  fSDUvdA,  measured 
in  coulombs.  Similarly  for  the  magnetic  induction  B, 
the  discharge  is  measured  in  maxwells. 

7.  Energy-Density.  Activity-Density.  Among  other 
scalar  products  of  importance  we  find  the  following.  If 
E  and  D  are  the  electric  intensity  in  volts/cm.  and  induction 
in  lines  at  a  point,  —  |$ED  is  the  energy-density  in  the 
field  at  the  point  in  joules/cc.  If  H  and  B,  likewise,  are  the 
magnetic  intensity  in  gilberts/cm.,  and  gausses,  respectively, 
—  2^#HB  is  the  energy  in  ergs.  If  J  is  the  electric  cur- 
rent-density in  amperes/cm.2,  —  S  •  E J  is  the  activity  in 
watts/cc.  If  G  is  the  magnetic  current-density  in  heavi- 
sides*/cm.2,  —  S  ■  H  G  is  the  activity  in  ergs/sec.  If  the 
field  varies  also,  the  electric  activity  is  —  >S-  E(J  +  D)  and 
the  magnetic  activity  —  $H(G  +  B). 

EXERCISES 

1.  An  insect  has  to  crawl  up  the  inside  of  a  hemispherical  bowl,  the 
coefficient  of  friction  being  1/3,  how  high  can  it  get? 

2.  The  force  of  gravity  may  be  expressed  in  the  form  a  =  —  mgk. 
Show  that  the  circulation  from  A  to  B  is  the  product  of  the  weight  by 
the  vertical  difference  of  level  of  A  and  B. 

3.  If  the  force  of  attraction  of  the  earth  is  <r  =  —  hUp/p2  show 
that  the  work  done  in  going  from  A  to  B  is 

hiTa-1  -  T0-1]. 

4.  The  magnetic  field  at  a  distance  a  from  the  central  axis  of  an 
infinite  straight  wire  carrying  a  current  of  electricity  of  /  amperes  is 

H  =  0.2ia-1(—  sin  di  +  cos  6j)     (i  andj  perpendicular  to  wire) 
and  the  differential  tangent  to  a  circle  of  radius  a  is   ( —  a  sin  6  i 
+  a  cos  9j)dd.     Show  that  the   gilbertage   is   0.2/  (02  —  0i)    gilberts, 
which  for  one  turn  is  OAirl. 

Prove  that  we  get  the  same  result  for  a  square  path. 

5.  The  permittivity  k  of  a  specimen  of  petroleum  is  2  [abfarad/cm.], 
and  on  a  small  sphere  is  a  charge  of  0.0001  coulomb.  The  value  of 
the  displacement  D  at  the  point  p  is  then 

D  =  9^2  UplTp2  [lineg] 

*  A  heaviside  is  a  magnetic  current  of  1  maxwell  per  second. 


132  VECTOR  CALCULUS 

What  is  the  discharge  through  an  equilateral  triangle  whose  corners 
are  each  4  cm.  from  the  origin,  the  plane  of  the  triangle  perpendicular 
to  the  field? 

6.  If  magnetic  inductivity  p.  is  1760  [henry/cm.]  and  a  magnetic 
field  is  given  by 

H  =  la  [gilbert/cm.], 

then  the  magnetic  induction  is 

B  =  7 -1760a  [gausses]. 

What  is  the  flux  through  a  circular  loop  of  radius  a  crossing  the  field 
at  an  angle  of  30°? 

7.  If  the  velocity  of  a  stream  is  given  by 

<r  =  24(cos  6  i  -f  sin  dj), 

what  is  the  discharge  per  second  through  a  portion  of  the  plane  whose 
equation  is  Sip  =  —  12  from 

d  =  10°        to        6  =  20°? 

8.  The  electric  induction  due  to  a  charge  at  the  origin  of  e  coulombs  is 

D  =  -  eUp/TPHir  [lines]. 

What  is  the  total  flux  of  induction  through  a  parallelepiped  whose 
center  is  the  origin? 

9.  The  magnetic  induction  due  to  a  magnetic  point  of  m  maxwells  is 

B  =  -  mUp/Tp2  [gausses]. 

What  is  the  total  flux  of  induction  through  a  sphere  whose  center  is 
the  point? 

10.  In  problem  8,  if  the  permittivity  is  2  =  k,  then  the  electric 
intensity 

E  =  rH>4r. 

What  is  the  amount  of  energy  enclosed  in  a  sphere  of  radius  3  cm.  and 
center  at  a  distance  from  the  origin  of  10  cm.? 

11.  In  problem  9,  if  the  inductivity  is  1760  and  the  magnetic  in- 
tensity is 

H  =  p~% 

how  much  energy  is  enclosed  in  a  box  2  cm.  each  way,  whose  center 
is  10  cm.  from  the  point  and  one  face  perpendicular  to  the  line  joining 
the  point  and  the  center? 

12.  If  the  current  in  a  wire  1  mm.  in  diameter  is  10  amperes  and 
the  drop  in  voltage  is  0.001  per  cm.,  what  is  the  activity? 


APPLICATIONS  133 

13.  If  there  is  a  leakage  of  10  heavisides  through  a  magnetic  area  of 
4  cm.2,  and  the  magnetic  field  is  5  gilberts/cm.,  what  is  the  activity? 

14.  Through  a  circular  spot  in  the  bottom  of  a  tank  which  is  kept 
level  full  of  water  there  is  a  leakage  of  100  cc.  per  second,  the  spot 
having  an  area  of  20  cm.2.  If  the  only  force  acting  is  gravity  what  is 
the  activity? 

15.  If  an  electric  wave  front  from  the  sun  has  in  its  plane  surface 
an  electric  intensity  of  10  volts  per  cm.,  and  a  magnetic  intensity  of 
0033  gilberts  per  cm.,  and  if  for  the  free  ether  or  for  air  y.  =  1  and 
k  =  £-10~20,  what  is  the  energy  per  cc.  at  the  wave  front?  (The 
average  energy  is  half  this  maximum  energy  and  is  according  to  Langley 
4.3  -10-5  ergs  per  cc.  per  sec.) 

16.  If  a  charge  of  e  coulombs  is  at  a  point  A  and  a  magnetic  point 
at  B  has  m  maxwells,  what  is  the  energy  per  cc.  at  P,  any  point  in  space, 
the  medium  being  air? 

8.  Geometric  Loci  in  Scalar  Equations. 
(1).  The  equation  of  the  sphere  may  be  written  in  each 
of  the  forms 

a/p  =  Kp[a, 

S(p  -  a)/(p  +  a)  =  0, 

S2a/(p  +  a)  =  1, 

S2p/(p  +  <*)  -  1, 

T(Sp/a  +  Vp/a)  =  1, 

Tip  -  ca)  m  T(cp  -  a), 
S{p  -  a) (a  -  »08  -  7)(Y  -  B)(S  -  p)  -  0, 

a2Sfiyp  +  j32Syap  +  y2Sa(3p  =  p2Sa(3y 
0  (p-aO2      (p-/3)2      (p-7)2     (P-5)2 

(p  -  a)2  0  (a  -  /3)2     (a  -  y)2     (a  -  5)2 

(p-/?)2     (/? -«)2  0  (/5-t)2     (0-S)2 

(P-T)2     (Y-«)2     (Y-0)2  0  (7-5) 

(p  -  5)2     (5  -  a)2     (5  -  /3)2      (5  -  7)2  0 


Interpret  each  form. 

(2).  The  equation  of  the  ellipsoid  may  be  written  in  the 
forms 

S2p/a  -  V2p/(3  =  1, 

where  a  is  not  parallel  to  ft 

T(p/y  +  Kpjb)  =  T(p/8  +  tfp/7), 

rOup  +  pX)=x2-/*2. 


134  VECTOR  CALCULUS 

The  planes 

a  p 

cut  the  ellipsoid  in  circular  sections  on  Tp  =  Tfi.  These 
are  the  cyclic  planes.  Tfi  is  the  mean  semi-axis,  Ufi  the 
axis  of  the  cylinder  of  revolution  circumscribing  the  ellip- 
soid, a  is  normal  to  the  plane  of  the  ellipse  of  contact  of 
the  cylinder  and  the  ellipsoid. 
In  the  second  form  let 

r1  -  -£,     7-1  =  -  £>      t2  =  n2  -  TJ, 

then  the  semi-axes  are 

a=rX+7>,        6=  ^~  TfX*>        c=T\-T». 
T(\  -  n) 

(3).  The  hyperboloid  of  two  sheets  is  S2p/a  +  F2p//3  =  1. 
(4).  The  hyperboloid  of  one  sheet  is  S2p/a  +  V2p/(3  =  —  1. 
(5) .  The  elliptic  paraboloid  of  revolution  is 

SplP+V2p/(3  =  0. 

(6).  The  elliptic  paraboloid  is  Sp/a  +  V2p/(3  =  0. 
(7).  The  hyperbolic  paraboloid  is  Sp/a  Sp/fi  =  Sp/y. 
(8).  The  torus  is 

T(±  bUarWap  -  p)  =  a, 

2bTVap  =  ±  (Tp2  +b2-  a2), 

4b2S2ap  =  4b2T2p  -  (T2p  +  b2  -  a2)2, 

Aa2T2p  -  4b2S2ap  =  (T2p  -  b2  +  a2)2, 

SU(p  -  «V  (a2  -  b2))l(p  +  cW  (a2  -  b2))  =  ±  b/a, 

p  =  ±  bJJoTWar  +  at/Y,         r  any  vector. 

(9).  Any  surface  is  given  by 

p  =  <p(u,  v). 


APPLICATIONS  135 

A  developable  is  given  by  p  =  <p(t)  +  ucp'it). 
(10).  A  cone  is  f(U[p  -  a])  =  0. 
The  quadric  cone  is  SapSfip  —  p2  =  0. 
The  cone  through  a,  (3,  y,  8,  e  is 

S-V(Va(3V8e)V(V(3yVep)V(Vy8Vpa)  =  0, 

which  is  Pascal's  theorem  on  conies. 

The  cones  of  revolution  through  X,  n,  v  are 

The  cones  of  revolution  which  touch  S\p  =  0,  Sfxp  =  0, 
Svp  =  0,  are 

The  cone  tangent  to  (p  —  a)2  +  c2  = 0  from  /?  is 

c2(p  -a-$)2=  V2(3(p  -  a). 

The  polar  plane  of  /3  is  £/3(p  —  a)  —  —  c2. 
The  cone  tangent  to 

a  p 

from  7  is 

(*i-F,S-1)(fl,J'-pi-0 

-( S^S^--  SV?V?--  lY=0. 
\     a     a  a      a  / 

The  cylinder  with  elements  parallel  to  y  is 

(s*i-f1-i)H-p?) 

_(s>sl-sv>vl)2  =  o. 

\     a     a  a     a) 


136  VECTOR  CALCULUS 

For  further  examples  consult  Joly :  Manual  of  Quater- 
nions. 

2.    The  Vector  of  Two  Vectors 
Notations,     If  a  and  /3  are  two  fields,  we  shall  call  V-a(3 
the  torque  of  the  two  fields. 

Va(3  =  Va(3       Hamilton,   Tait,   Joly,   Heaviside,   Foppl, 
Ferraris,  Carvallo. 
cqS  Grassman,  Jahnke,  Fehr. 

aX  0    Gibbs,  Wilson,  Fischer,  Jaumann,  Jung. 
[a,  /3]      Lorentz,  Gans,  Bucherer,  Abraham,  Timer- 
ding. 
[a  |  /?]       Caspary . 

a  A  j3    Burali-Forti,  Marcolongo,  Jung. 
aj8  Heun. 

Sin  a/3  Macfarlane. 

Iaccb        Peano. 
1.  Lines.     The  condition  that  two  lines  be  parallel  is  that 
Vafi  =  0.     Therefore  the  equation  of  the  line  through  the 
origin  in  the  direction  a  is  Vap  —  0. 

The  line  through  0  parallel  to  a  is  Va(p  —  fi)  =  0  or 
Vap  =  Va(3  =  y.  The  perpendicular  from  5  on  the  line 
Vap  =  7  is 

—  a~lVab  +  a~ly. 

The  line  of  intersection  of  the  planes,  S\p  =  a,  S^p  =  b,  is 
VpV\fx  =  a/x  —  6X.  If  we  have  lines  Vpa  —  y  and  Vp&  =  8 
then  a  vector  from  a  point  on  the  first  to  a  point  on  the 
second  is  5/3"1  —  7a-1  +  #/3  —  ya.  If  now  the  lines  in- 
tersect then  we  can  choose  x  and  y  so  that  this  vector  will 
vanish,  corresponding  to  the  two  coincident  points,  and 
thus 

S{bp~l  -  ya~l)$a  =  0  =  S8a  +  Syp. 


APPLICATIONS  137 

If  we  resolve  the  vector  joining  the  two  points  parallel  and 
perpendicular  to  Vaft  we  have* 

5/3-1  —  ya~l  +  xfi  —  ya 

=  •  (Va^S  •  VaP(bprl  -  yoT1  +  zp  -  ya) 

=  -(VaP)-\S5(x+  Spy) 


L  a     Fa/3  Fa/3  J 

L         Va0         P     Vap] 

-  «-*  f-  SaPS  ^~  +  a2S  JL  1 
Vap  Vap] 


Hence  the  vector  perpendicular  from  the  first  line  to  the 
second  is 

-  (Vafl-KStct  +  Spy) 

and  vectors  to  the  intersections  of  this  perpendicular  with 
the  first  and  second  lines  are  respectively 


and 


ya  x  —  a  1  \  8  ' — ^— 

L  Va(3        J 


*  Note  that 

(Va0)-lV(Vu0)(z0  -  ya)  =  xp  -  ya 
(y«jS)-1F-7a/3(5J3-1  -  ya~l)  =  (Vc0)-l(-  a'^Sfiya  -  (rlS<*&) 


Va,(-^S^+p-S^) 


10 


138  VECTOR  CALCULUS 

The  projections  of  the  vectors  a,  y  on  any  three  rectangular 
axes  give  the  Pluecker  coordinates  of  the  line.  For  applica- 
tions to  linear  complexes,  etc.,  see  Joly:  Manual,  p.  40, 
Guiot:  Le  Calcul  Vectoriel  et  ses  applications. 

2.  Congruence.  The  differential  equation  of  a  curve  or 
set  of  curves  forming  a  congruence  whose  tangents  have 
given  directions  cr,  that  is,  the  vector  lines  of  a  vector  field  <r, 
is  given  by 

Vdpa  =  0 

or  its  equivalent  equation 

dp  =  adt. 

3.  Moment.  The  moment  of  the  force  /3  with  respect 
to  a  point  whose  vector  from  an  origin  on  the  line  of  @  is 
a,  is  —  Fa/3.  If  the  point  is  the  origin  and  the  vector  to 
some  point  in  the  line  of  application  of  the  force  is  a,  then 
the  moment  with  respect  to  the  origin  is  Vafi.  If  the 
point  is  on  the  line  of  application  the  moment  obviously 
vanishes.  If  several  forces  have  a  common  plane  then 
the  moments  as  to  a  point  in  the  plane  will  have  a  common 
unit  vector,  the  normal  to  the  plane.  If  several  forces  are 
normal  to  the  same  plane,  their  points  of  application  in 
the  plane  given  by  ft,  ft,  ft,  •  •  • ,  their  values  being  a\a> 
a2a,  asa,  •  •  • ,  then  the  moments  are 

F(aift  +  a2ft  +  a3ft  +  •  •  •)«        [dyne  cm.]. 
If  we  set 

«ift  +  02ft  +  03ft  +  •  •  •  =  ftai  +  a2+  az+  •  •  •)/ 

then  /3  is  the  vector  to  the  mean  point  of  application,  which, 
in  case  the  forces  are  the  attractions  of  the  earth  upon  a 
set  of  weighted  points,  is  called  the  center  of  gravity.  If 
ai  +  #2  +  a3  +  •  •  •  =  0,  we  cannot  make  this  substitution. 


APPLICATIONS  139 

4.  Couple.  A  couple  consists  of  two  forces  of  equal 
magnitude,  opposite  directions  and  different  lines  of  action. 
In  such  case  the  mean  point  becomes  illusory  and  the  sum 
of  the  moments  for  any  point  from  which  vectors  to  points 
on  the  lines  of  action  of  the  forces  are  ah  a2  respectively,  is 

V{ax  -  a2)P. 

But  a\  —  a2  is  a  vector  from  one  line  of  action  to  the  other, 
and  this  sum  of  the  moments  is  called  the  moment  of  the 
couple.  It  is  evidently  unchanged  if  the  tensor  of  /?  is 
increased  and  that  of  a\  —  a2  decreased  in  the  same  ratio, 
or  vice  versa. 

5.  Moment  of  Momentum.  If  the  velocity  of  a  moving 
mass  m  is  a  cm./sec,  then  the  momentum  of  the  mass  is 
defined  to  be  ma  gr.  cm./sec.  The  vector  to  the  mass 
being  p,  the  moment  of  momentum  of  the  mass  is  defined 
to  be 

Vpma  =  mVpa  [gm.  cm.2/sec.]. 

6.  Electric  Intensity.  If  a  medium  is  moving  in  a  mag- 
netic field  of  density  B  gausses,  with  a  velocity  a  cm./sec, 
then  there  will  be  set  up  in  the  medium  an  electromotive 
intensity  E  of  value 

E=Fo-B-10~8        [volts/centimeter]. 

For  any  path  the  volts  will  be 

-  fSdPE=  +  fSdpBa-10-8. 

If  this  be  integrated  around  any  complete  circuit  we  shall 
arrive  at  the  difference  in  electromotive  force  at  the  ends 
of  the  circuit. 

7.  Magnetic  Intensity.  If  a  magnetic  medium  is  moving 
in  an  induction  field  of  D  lines,  with  a  velocity  a,  then  there 
will  be  produced  in  the  medium  at  every  point  a  magnetic 


140  VECTOR  CALCULUS 

intensity  field 

H  =  OAwVDa  [gilberts/cm.]. 

For  any  path  the  gilbertage  will  be  OAirf  SdpaD. 

8.  Moving  Electric  Field.  If  an  electric  field  of  induc- 
tion, of  value  D  lines,  is  moving  with  a  velocity  a,  then 
there  will  be  produced  in  the  medium  at  the  point  a  mag- 
netic field  of  intensity  H  gilberts/cm.  where 

H  m  OAirVaD. 

For  a  moving  electron  with  charge  e,  this  will  be 
—  (eUp/4:irTp2).  For  a  continuous  stream  of  electrons 
along  a  path  we  would  have 

the  point  being  the  origin. 

9.  Moving  Magnetic  Field.  If  a  magnetic  field  of  in- 
duction of  value  B  gausses  is  moving  with  a  velocity  cr, 
it  will  produce  at  any  given  point  in  space  an  electric 
intensity  E  =  V  -  BolO-8  volts  per  centimeter. 

10.  Torque.  If  a  particle  of  length  dp  is  in  a  field  of 
intensity  <r  which  tends  to  turn  the  particle  along  the  lines 
of  force,  then  the  torque  produced  by  the  field  upon  the 
element  is 

V-dpa. 

If  a  line  runs  from  A  to  B,  the  total  torque  is 

//  V-dpe. 
For  instance  if  dp,  or  in  case  of  a  non-uniform  distribution 
cdp,  is  the  strength  in  magnetic  units,  maxwells,  of  a  wire 
magnet  from  A  to  B,  in  a  field  a,  then 

fIV-dpa         or  f/V-cdpa 

is  the  torque  of  the  field  upon  the  magnet. 


APPLICATIONS  141 

11.  Poynting  Vector.  An  electric  intensity  E  volts/cm. 
and  magnetic  intensity  H  gilberts/cm.  at  a  point  in  space 
are  accompanied  by  a  flux  of  energy  per  cm.2  R,  given  by 
the  formula 

4xR  =  — —  [ergs/cm.2  sec.]. 

This  is  the  Poynting  vector. 

12.  Force  Density.  The  force  density  in  dynes/cc.  of 
a  field  of  electric  induction  on  a  magnetic  current  is  given  by, 

F  =  4ttFDG  :  10  [dynes/cc], 

where  D  is  the  density  in  lines  of  electric  displacement 
G  is  the  magnetic  current  density  in  heavisides  per  cm.2. 
If  the  negative  of  F  is  considered  we  have  the  force  per  cc. 
required  to  hold  a  magnetic  current  in  an  electrostatic 
field  of  density  D. 

The  force  density  in  dynes/cc.  of  a  field  of  magnetic 
induction  on  a  conductor  carrying  an  electric  current  is 

F-ijr.il. 

A  single  moving  charge  e  with  velocity  a  will  give 
F  =AweVaiJiVaD. 

13.  Momentum  of  Field.  The  field  momentum  at  a 
point  where  the  electric  induction  is  D  lines  and  magnetic 
induction  B  gausses  is  T  =  3-109  V-  DB  [gm.  cm./sec.].  If 
the  magnetic  induction  is  due  to  a  moving  electric  field  then 
T  =  0.047rF-  D/jlVDct,  and  if  the  electric  induction  is  due 
to  a  moving  magnetic  field, 

T  = VB/cVaB. 

47T-3-1010 


142  vector  calculus 

3.    The  Scalar  of  Three  Vectors 

1.  Area  and  Pressure.  If  we  consider  two  differential 
vectors  from  the  point  P,  say  dip,  d2p,  then  the  vector  area 
of  the  parallelogram  they  form  is  Vdipd2p.  If  then  we  have 
a  distribution  of  an  areal  character,  such  as  pressure  per 
square  centimeter,  /3,  the  pressure  normal  to  the  differential 
area  will  be  in  magnitude 

—  S(3dipd2p. 

The  vector  Vdipd2p  may  be  represented  by  dp  or  JJvdA. 
The  vector  pressure  normal  to  the  surface  will  be 

UpS(3dipd2p. 

There  will  also  be  a  tangential  pressure  or  shear,  which  is 
the  other  component  of  /3. 

2.  Flux.  If  j8  is  any  vector  distribution  the  expression 
—  S($d\pd2p  is  often  called  the  flux  of  /?  through  the  area 
Vd\pd2p.  It  is  to  be  noted  however  that  the  dimensions 
of  the  result  in  physical  units  must  be  carefully  considered. 
Thus  the  flux  of  magnetic  intensity  is  of  dimensions  that 
do  not  correspond  to  any  magnetic  quantity. 

3.  Flow.  If  /3  is  the  velocity  of  a  fluid  in  cm./sec,  then 
the  volume  passing  through  the  differential  area  per  second 
is 

—  Sfid\pd2p  [cc./sec.]. 

4.  Energy  Flux.  The  dimensions  of  the  Poynting  energy 
flow  R  show  that  it  is  the  current  of  energy  per  second  across 
a  cm.2,  hence  the  total  flow  per  second  through  an  area  is 

-SRd^p--8-™™^    [ergs/sec] 

In  the  case  of  a  straight  conductor  carrying  a  current  of 
electricity,  we  have  at  a  distance  a  from  the  wire  in  a 


APPLICATIONS  143 

direction  at  right  angles  to  the  wire  directly  away  from  it 
the  value 

T  R=  (4ir)-1108JS;(0.2Ja-1). 

Consequently  if  we  consider  one  centimeter  of  wire  in 
length  and  the  circumference  of  the  circle  of  radius  a  we 
shall  have  a  flux  of  energy  for  the  centimeter  equal  to 

J(ft-«0  [jouks]. 

This  is  the  usual  J2R  of  a  wire  and  is  represented  by  heat. 

5.  Activity.  For  a  moving  conductor  we  have  already 
expressed  the  vector  E,  and  as  the  current  density  J  can 
be  computed  from  the  intensity  of  the  field  (J  =  k  E)  we 
have  then  for  the  expression  of  the  activity  in  watts  per 
cubic  centimeter  of  conductor 

A=  -SaBhO-8=  -S(V(rB)k(VaB)-lO-ie     [watts]. 

Likewise  in  the  case  of  the  magnetomotive  force  due  to 
motion  and  the  magnetic  current  G  =  IH  we  have  for  the 
activity  per  cubic  centimeter  of  circuit 

A=-  SDaQ  =  -  S-(VDa)l(VD(r)-10-7     [watts]. 

6.  Volts.  The  total  electromotive  difference  between 
two  points  in  a  conductor  is  the  line-integral  along  the 
conductor 

-  fSdpaBlCr*  [volts]. 

7.  Gilberts.  The  total  magnetomotive  difference  be- 
tween two  points  along  a  certain  path  is  the  line-integral 

—  AirfSdpDo-  [gilberts]. 

4.    Vector  of  Three  Vectors 
1.  Stress.     We  find  with  no  difficulty  the  equations 

V-a(Ua±  Uy)y  =  ±  TyTa(Ua±  Uy) 
and 

V-a(Vay)y  =  —  Say  -V -ay. 


144  VECTOR  CALCULUS 

If  now  we  have  a  state  of  stress  in  a  medium,  given  by  its 
three  principal  stresses  in  the  form 

0i  =  g  —  7V  dynes/cm.2  normal  to  the  plane  orthogonal 

to  U(U\+  Un), 
92  =  g  —  S\fx  dynes/cm.2  normal  to  the  plane  orthogonal 

to  UV\fi, 
gz  =  g  +  T\p  dynes/cm.2  normal  to  the  plane  orthogonal 
to  U{U\  -  Un), 

gi  <  gi  <  gz, 

then  the  stress  across  the  plane  normal  to  /?  is 

V\fo  +  0. 

If  the  scalars  git  g%,  gz  are  dielectric  constants  in  three 
directions  (trirectangular)  properly  chosen,  then  the  dis- 
placement is 

D  =  FXE/x  +  gE. 

If  the  scalars  are  magnetic  permeability  constants, 

B  =  V\Hfi  +  gW. 
If  the  scalars  are  coefficients  of  dilatation,  then  0  becomes 

(T--  VWp+gp. 

If  the  scalars  are  elasticity  constants  of  the  ether,  then 
according  to  Fresnel's  theory,  the  force  on  the  ether  is, 
for  the  ether  displacement  ft  .  . 

V\fo  +  gp. 

If  the  scalars  are  thermoelectric  constants  in  a  crystal, 
then 

D  =  FXQm  +  gQ.        where  Q  is  the  flow  of  heat. 

If  g  =  0  the  scalars  are  TV,  -  TV,  -  SXfi.  If  V\fi  =  0, 
the  scalars  are  7V>  —  TV>  T\p,  that  is,  practically  —  t 
along  X  and  +  t  in  all  directions  perpendicular  to  X. 


CHAPTER   VIII 

DIFFERENTIALS   AND   INTEGRALS 
1.      DlFFEKENTIATION   AS   TO   A   SCALAR   PARAMETER 

1.  Differential  of  p.  If  the  vector  p  depends  upon  the 
scalar  parameter  t,  say 

p  =  <p(t), 

then  for  two  values  of  t  which  are  supposed  to  be  in  the 
range  of  possible  values  for  t 

Pi  —  Pi  =  <p(h)  —  <p(ti)  t 
ti  —  t\  t%  —  ^1 

If  now  we  suppose  that  U  <  h  <  t2  and  that  h  and  t2  can 
independently  approach  the  limit,  t0,  then  we  shall  call 
the  limit  of  the  fraction  above,  if  there  be  such  a  limit,  the 
right-hand  derivative  of  p  as  to  t,  at  t0,  and  if  t2  <  h  <  t0, 
we  shall  call  the  limit  the  left-hand  derivative  of  p  as  to 
t  at  t0.  In  case  these  both  exist  and  are  equal,  and  if  p 
has  a  value  for  t0  which  is  the  limit  of  the  two  values  of 
<p(ti),  then  we  shall  say  that  p  is  a  continuous  function  of 
t  at  t0  and  has  a  derivative  as  to  t  at  to. 

There  is  no  essential  difference  analytically  between  the 
function  <p  and  the  ordinary  functions  of  a  single  real 
variable,  and  we  will  assume  the  ordinary  theory  as  known. 

It  is  evident  that  for  different  values  of  t  we  may  con- 
sider the  locus  of  P  which  will  be  a  continuous  curve. 
Since  p2  —  pi  is  a  chord  of  the  curve  the  limit  above  will  give 
a  vector  along  the  tangent  of  the  curve.  Further  the  tensor 
of  the  derivative,  Tp'  =  T(p'{t)y  is  the  derivative  of  the 
length  of  the  arc  as  to  the  parameter  t.  If  the  arc  s  is  the 
parameter  then  the  vector  p'  is  a  unit  vector. 

145 


146  VECTOR  CALCULUS 

EXAMPLES 

(1)  The  circle 

p  =  a  cos  0  -f  0  sin  0,  To:  =  Fft         Sa0  =  0, 

p'  =  —  a  sin  0  +  0  cos  0. 

(2)  The  helix 

p  =  a  cos  0  +  0  sin  0  +  70, 
p'  =  —  a  sin  0  +  0  cos  0+7. 

(3)  The  conic 

_  a*2  +  20<  +  7 
P       at2  +2U+C  ' 

Multiplying  out,  t2(a  -  ap)  +  2*(0  -  bp)  +  (7  -  cp)  =  0 
for  all  values  of  £.     For  2  =  0,  p  =  7/c,  and  for  t  =  00 , 
p  =  a/a,  hence  the  curve  goes  through  a/a  and  7/c. 
We  have 

rfp/d<  =  [t2(ba  —  a0)  +  <(ca  —  07)  +  (c0  —  by)]  times  scalar. 

Hence  for  t  =  0,  the  direction  of  the  tangent  is  0/6  —  7/c 
at  7/c,  for  t  =  00,  the  direction  of  the  tangent  is 
0/6  —  a/a  at  a/a.  Since  these  vectors  both  run  from  the 
points  of  tangency  to  the  point  0/6,  the  curve  is  a  conic, 
tangent  to  the  lines  through  0/6  and  the  two  points  a/a 
and  7/c,  at  these  two  points.  If  the  origin  is  taken  at 
0/6,  so  that  p  =  w  +  0/6,  and  if  a!  =  a/a  —  0/6,  7'  =  7/c 
-  0/6,  then 

at\a!  -  tt)  -  26/tt  +  c(yf  -  w)  =  0 

is  the  equation  of  the  curve. 

If  now  we  let  w  run  along  the  diagonal  of  the  parallelo- 
gram whose  two  sides  are  a'y'  so  that  tt  =  x(a!  +  y'),  then 
substituting  we  have 

at2x  +  2btx  -  c(l  -  x)  =  0, 

at2(l  -  x)  -  2btx  -  ex  =  0. 


DIFFERENTIALS  147 

From  these  equations  we  have 

t2  =  c/a 
and 

x  =  Vac/2(Vac±  b). 

These  values  of  x  give  us  the  two  points  in  which  the 
diagonal  in  question  cuts  the  curve.  The  middle  point 
between  these  two  is 

Referred  to  the  original  origin  this  gives  for  the  center 

,      ,,       ca  -  2b(3  +  ay 

k=  r  +  p  b  =  — — =£—  • 

2(ac  —  b2) 

If  we  calculate  the  point  on  the  curve  for 

bh  +  e 


ah+  b 


we  shall  find  that  for  the  points  p2,  pi  we  have  J(p2  +  Pi) 
=  k,  so  that  k  is  the  center  of  the  curve  and  diametrically 
opposite  points  have  parameters 

h  and  t2  =  — — r-x  > 
ati  ~t  o 

an  involutory  substitution.  If  ac  =  b2,  k  becomes  co  ex- 
cept when  also  the  numerator  =  0.  [Joly,  Manual,  Chap. 
VII,  art.  48.] 

In  general  the  equation  of  the  tangent  of  any  curve  is 

IT  =  p  +  Xp'. 

We  may  also  find  the  derivatives  of  functions  of  p,  when 
p  =  (p(t),  by  substituting  the  value  of  p  in  the  expression 
and  differentiating  as  before.     Thus 

let  p  =  a  cos  6  +  P  sin  6        where  Ta  #  Tp. 


148  VECTOR  CALCULUS 

Then 

Tp  =  V  [-  a2  cos2  6  -  2Sap  sin  6  cos  6  -  02  sin2  6], 

We  may  then  find  the  stationary  values  of  Tp  in  the  manner 
usual  for  any  function.     Thus  differentiating  after  squaring 

a2  sin  26  -  2Sa(3  cos  26  -  fi2  sin  26  =  0, 
tan  26  -  2Sap/(a2  -  /32). 

2.  Frenet-Serret  Formulae.  Since  the  arc  is  essentially 
the  natural  parameter  of  a  curve  we  will  suppose  now  that 
p  is  expressed  in  terms  of  s,  and  accents  will  mean  only 
differentiation  as  to  s.    Then  both 

p        and        p  +  dsp' 

are  points  upon  the  curve. 

The  derivative  of  the  latter  gives  p'  +  dsp",  which  is  also 
a  unit  vector  since  the  parameter  is  s.  Thus  the  change  in 
a  unit  vector  along  the  tangent  is  dsp",  and  since  this 
vector  is  a  chord  of  a  unit  circle  its  limiting  direction  is 
perpendicular  to  p',  and  its  quotient  by  ds  has  a  length  whose 
limit  is  the  rate  of  change  of  the  angle  in  the  osculating  plane 
of  the  tangent  and  a  fixed  direction  in  that  plane  which 
turns  with  the  plane.  That  is  to  say,  p"  in  direction  is 
along  the  principal  normal  of  the  curve  on  the  concave  side, 
and  in  magnitude  is  the  curmture  of  the  curve,  which  we 
shall  indicate  by  the  notation 

Unit  tangent  is     a  =  p', 

Unit  normal  is      |9  =  Up",  curvature  is    Ci  =  Tp", 

Unit  binormal  is  y  =  Va(3,  so  that  Ciy  =  Vp'p". 

The  rate  of  angular  turn  of  the  osculating  plane  per  centi- 
meter of  arc  is  found  by  differentiating  the  unit  normal  of 
the  plane.     Thus  we  have 

Ti  =  cf2hW  -  Fp'p"-c2].' 


DIFFERENTIALS  149 

But  d2  =  T2p"  =  -  Sp"p"  and  therefore  Clc2  =  -  SP"pf". 
Substituting  for  c2  we  have 

71  -  cr3[-  Sp"p"Vp'pf"  +  SpV'Wl 
=  crz[Vp,Vp"Vp,,,p"] 

=  crWaVc1(3Vp'"c1p 

=  crW-aPVp"^  =  crlVyVp'"p  =  crlpSyp"' 

=  -  «lft 

where  «i  is  written  for  the  negative  tensor  of  71  and  is  the 
tortuosity.     It  may  also  be  written  in  the  form 

Again  since  /?  =  ya  we  have  at  once  the  relations 

j3i  =  7i«  +  7«i  =  «i7  ~~  C\a. 

Thus  we  have  proved  Frenet's  formulae  for  any  curve 

«i  =  erf,        ft  =  ai7  —  ci«,        71  =  —  a$. 

It  is  obvious  now  that  we  may  express  derivatives  of  any 
order  in  terms  of  a,  ft  y,  and  Oi,  Ci,  and  the  derivatives  of 
ai  and  Ci. 

For  instance  we  have 

Pi  =   OL,  p2  =  fci, 

Pa  =  ftci  +  Pc2  =  fe  +  (701  —  aci)ci, 
Pi  =  0c3  +  2{yai  —  aci)c2  +  (ya2  —  ac2)ci 

-  ^(ai2  +  Cl2)Cl. 

The  vector  w  =  aai  +  7C1  is  useful,  for  if  77  represents  in 
turn  each  one  of  the  vectors  a,  /3,  7,  then  771  =  Fa^     It  is 
the  vector  along  the  rectifying  line  through  the  point. 
The  centre  of  absolute  curvature  k  is  given  by 

K  =  p  -  lip"  m  p  +  Pld. 


150  VECTOR  CALCULUS 

The  centre  of  spherical  curvature  is  given  by 

a  =  k  +  yd/da  •  c{~1  =  k  —  yc2/aiCi2. 

The  polar  line  is  the  line  through  K  in  the  direction  of  7. 
It  is  the  ultimate  intersection  of  the  normal  planes. 

3.  Developable s.  If  we  desire  to  study  certain  de- 
velopables  belonging  to  the  curve,  a  developable  being  the 
locus  of  intersections  of  a  succession  of  planes,  we  proceed 
thus.  The  equation  of  a  plane  being  S(w  —  p)rj  =  0, 
where  t  is  the  vector  to  a  variable  point  of  the  plane,  and 
p  is  a  point  on  the  curve,  while  rj  is  any  vector  belonging 
to  the  curve,  then  the  consecutive  plane  is 

S(t  -  p)f)  +  ds'd/dsS(w  -  p)r)  =  0. 

The  intersection  of  this  and  the  preceding  plane  is  the  line 
whose  equation  is 

7r  =  p  +  (—  r)Sar)  +  t)lVr}r}i. 

This  line  lies  wholly  upon  the  developable.  If  we  find  a 
secOnd  consecutive  plane  the  intersection  of  all  three  is  a 
point  upon  the  cuspidal  edge  of  the  developable,  which  is 
also  the  locus  of  tangents  of  the  cuspidal  edge.  This  vector 
is 
tv  =  p  +  (VwySar}  +  2Vr)7]iSar)i  +  Vr}7}iS^rjCi)/ST]r}ir]2' 

By  substituting  respectively  for  77,  a,  ft  7,  we  arrive  at  the 
polar  developable,  the  rectifying  developable,  the  tangent- 
line  developable. 

EXAMPLE 
Perform  the  substitutions  mentioned. 

4.  Trajectories.  If  a  curve  be  looked  upon  as  the  path 
of  a  moving  point,  that  is,  as  a  trajectory,  then  the  param- 
eter becomes  the  time.  In  this  case  we  find  that  (if 
p  =  dp/dt,  etc.)  the  velocity  is  p  =  av,  the  acceleration  is 


DIFFERENTIALS  151 


p  =  ficiv2  +  av.  The  first  term  is  the  acceleration  normal 
to  the  curve,  the  centrifugal  force,  the  second  term  is  the 
tangential  acceleration.  In  case  a  particle  is  forced  to 
describe  a  curve,  the  pressure  upon  the  curve  is  given  by 
(3civ2.  There  will  be  a  second  acceleration,  p  =  a(v  —  wi2) 
+  (3(2cii  +  c2v)  +  yaiCiV.  The  last  term  represents  a 
tendency  per  gram  to  draw  the  particle  out  of  the  osculating 
plane,  that  is,  to  rotate  the  plane  of  the  orbit. 

5.  Expansion  for  p.  If  we  take  a  point  on  the  curve 
as  origin,  we  may  express  p  in  the  form 

p  =  sa  +  %cis2(3  —  %s*(ci2a  —  c2/3  —  cmy) 

—  ^4(3c2cia:  ~~  £IC3  ~~  c*  —  Clttl2l  ~~  T[2c2ai  +  da2]) 

EXERCISES 

1.  Every  curve  whose  two  curvatures  are  always  in  a  constant  ratio 
is  a  cylindrical  helix. 

2.  The  straight  line  is  the  only  real  curve  of  zero  curvature  every- 
where. 

3.  If  the  principal  normals  of  a  curve  are  everywhere  parallel  to  a 
fixed  plane  it  is  a  cylindrical  helix. 

4.  The  curve  for  which 

Ci  =  1/ms,        ai  =  1/ns, 
is  a  helix  on  a  circular  cone,  which  cuts  the  elements  of  the  cone  under 
a  constant  angle. 

5.  The  principal  normal  to  a  curve  is  normal  to  the  locus  of  the 
centers  of  curvature  at  points  where  Ci  is  a  maximum  or  minimum. 

6.  Show  that  if  a  curve  lies  upon  a  sphere,  then 

cr1  =  A  cos  a  +  B  sin  a  =  C  cos  (a  +  e),  A,  B,  C,  e  are  constants. 
The  converse  is  also  true. 

7.  The  binormals  of  a  curve  do  not  generate  the  tangent  surface  of 
a  curve. 

8.  Find  the  conditions  that  the  unit  vectors  of  the  moving  trihedral 
afiy  of  a  given  curve  remain  at  fixed  angles  to  the  unit  vectors  of  the 
moving  trihedral  of  another  given  curve. 

Two  Parameters 
6.  Surfaces.     If  the  variable  vector  p  depends  upon  two 
arbitrary  parameters  it  will  terminate  upon  a  surface  of 


152  VECTOR  CALCULUS 

some  kind.  For  instance  if  p  =  <p(u,  v),  then  we  may 
write  for  the  total  differential  of  p 

dp  =  dud/du(<p)  -f-  dvd/dv((p)  =  du<pu  +  dv<pv. 

We  find  then 

Fdp  =  £dw2  +  2Fdudv  +  GW, 
where 

E  =  —  ^tt2,        F  =  —  S<pu<pv,        G  =  —  ^t,2. 

We  have  thus  two  differentials  of  p,  one  for  »  =  constant, 
one  for  u  =  constant,  which  will  be  tangent  to  the  para- 
metric curves  upon  the  surface  given  by  these  equations, 
and  may  be  designated  by 

pidu,        p2dv. 

The  normal  becomes  then 

v  =  vPlp2,        Tv  =  V  (EG  -  F2)  =  H. 

For  certain  points  or  lines  v  may  become  indeterminate, 
the  points  or  lines  being  then  singular  points  or  singular 
lines. 

7.  Curvatures.     If   we  consider  the   point  p  and   the 
point  p  +  dupi  -f-  dvp2  the  two  normals  will  be 

v    and     v  +  duV(pnp2  +  P1P12) 

-f-  dvV(pi2p2  +  P1P22)  +  •  •  • 
which  may  be  written 

v    and     v  +  dv. 

The  equations  of  these  lines  are 

V(w  -  P)v  =0,         V(w-  p-  dP)(v  +  dv)  =  0. 

They  intersect  if 

Sdpvdv  =  0. 

Points  for  which  this  equation  holds  lie  upon  a  line  of 


DIFFERENTIALS  153 

curvature  so  that  this  is  the  differential  equation  of  such 
lines.     If  we  expand  the  total  differentials  we  have 

du2Spivi>i  +  2dudvS(piw2  +  Pivv\)  +  dv2Sp2w2  =  0. 

We  may  also  write  the  equation  in  the  form 

dp  +  xv  +  ydv  =  0  =  pidu  -\-  p2dv  +  xv  +  yv\du  +  yv2dv. 

Multiply  by  (pi  +  yv\){p2  +  yv2)  and  take  the  scalar  part 
of  the  product,  giving 

S(pi  +  yvi)(pi  +  P2#>  =  o 

=  y2Svviv2  +  2ySv{piv2  +  ^ip2)  +  ^2. 

The  ultimate  intersection  of  the  two  normals  is  given  by 

t  =  p  +  dp  +  yv  +  y<&>, 

that  is  by  yv.  Hence  we  solve  for  yTv,  giving  two  values 
R  and  Rf  which  are  the  principal  radii  of  curvature  at  the 
point.  The  product  and  the  sum  of  the  roots  are  re- 
spectively 

RR'  =  yy'Tv2  -  Tv%-  Sw1v2), 
R  +  R'  =  —  2TvSv(piv2  -\-  vip2)/Swiv2. 

The  reciprocal  of  the  first,  and  one-half  the  second  divided 
by  the  first,  that  is, 

—  Spvivt/v4        and         Sv(piv2  +  vip2)/Tv*, 

are  the  absolute  curvature  and  the  mean  curvature  of  the 
surface  at  the  point. 

The  equation  of  the  lines  of  curvature  may  be  also  written 

vSdpvdv  =  0  =  V-VdpVvdv  =  VdpV(dv/vv)  =  VdpdUv. 

Hence  the  direction  of  dUv  is  that  of  a  line  of  curvature, 
when  du  and  dv  are  chosen  so  that  dp  follows  the  line  of 
curvature.     That  is,  along  a  line  of  curvature  the  change 
li 


154  VECTOR  CALCULUS 

in  the  direction  of  the  unit  normal  is  parallel  to  the  line 
of  curvature. 

When  the  mean  curvature  vanishes  the  surface  is  a 
minimal  surface,  the  kind  of  surface  that  a  soapfilm  will 
take  when  it  extends  from  one  curve  to  another  and  the 
pressures  on  the  two  sides  are  equal.  The  pressure  indeed 
is  the  product  of  the  surface  tension  and  twice  the  mean 
curvature,  so  that  if  the  resultant  pressure  is  zero,  the 
mean  curvature  must  vanish.  If  the  radii  are  equal,  as  in 
a  sphere,  then  the  resultant  pressure  will  be  twice  the 
surface  tension  divided  by  the  radius,  for  each  surface  of  the 
film,  giving  difference  of  pressure  and  air  pressure  =  4 
times  surface  tension/radius.  The  difference  of  pressure 
is  thus  for  a  sphere  of  4  cm.  radius  equal  to  the  surface 
tension,  that  is,  27.45  dynes  per  cm. 

When  a  surface  is  developable  the  absolute  curvature  is 
zero,  and  conversely.  Surfaces  are  said  to  have  positive 
or  negative  curvature  according  as  the  absolute  curvature 
is  positive  or  negative. 

EXERCISES 

1.  The  differential  equation  of  spheres  is 

Vp(p  -  a)  =  0. 

2.  The  differential  equations  of  cylinders  and  cones  are  respectively 

Sva  =  0,         Sv(p  -  a)  =  0. 

3.  The  differential  equation  of  a  surface  of  revolution  is 

Sapv  =  0. 

4.  Why  is  the  center  of  spherical  curvature  of  a  spherical  curve  not 
of  necessity  the  center  of  the  sphere? 

5.  Show  how  to  find  the  vector  to  an  umbilicus  (the  radii  of  curvature 
are  equal  at  an  umbilicus). 

6.  The  differential  equation  of  surfaces  generated  by  lines  that  are 
perpendicular  to  the  fixed  line  a  is 

SVav<pVocv  =  0, 

where  <p  is  a  linear  function. 


DIFFERENTIALS  155 

7.  The  differential  equation  of  surfaces  generated  by  lines  that  meet 
the  fixed  line  V(p  —  (3)  a  =  0  is 

SVvV{p  -  P)a<p{VVV(p  -  0)a)  =  0. 

8.  The  differential  equation  of  surfaces  generated  by  equal  and 
similarly  situated  ellipses  is 

SV(Va&-p)v(VYa0-p)  =  0. 

9.  Show  that  the  catenoid 

p  =  xi  +  a  cosh  x/a(cos  8j  +  sin  6k) 

is  a  minimal  surface,  and  that  the  two  radii  are  db  Tv,  the  normal  which 
is  drawn  from  the  point  to  the  axis. 

2.    Differentiation  as  to  a  Vector 
1.  Definition.     Let  q  =  /(p)  be  a  function  of  p,  either 
scalar,  vector,  or  quaternion.     Let  p  be  changed  to  p  +  dt  •  a 
where  a  is  a  unit  vector,  then  the  change  in  q  is  given  by 

dq=  q'  -  q  =  f{p+  dt-a)  -  /(p), 
and 

dq/dt  =  Lim  [/(p  +  dta)  -  f(p)]/dt 

as  dt  decreases.     If  we  consider  only  the  terms  in  first 
order  of  the  infinitesimal  scalar  dt  we  can  write 

dq  =  dtf(p,  a) 

in  which  a  will  enter  only  linearly. 

In  a  linear  function  of  a  however  we  can  introduce  the 
multiplier  into  every  term  in  a  and  write  dta  =  dp,  so  that 
we  have  dq  a  linear  function  of  dp, 

dq  =  f'(p,  dp). 

It  needs  to  be  noted  that  the  vector  a  is  a  function  of  the  variable  dt, 
although  a  unit  vector.  The  differential  of  q  is  of  course  a  function  of 
the  direction  of  dp  in  general,  but  the  direction  may  be  arbitrary,  or  be 
a  function  of  the  variable  vector  p.  It  may  very  well  happen  that  the 
limit  obtained  above  may  be  different  for  a  given  function  /  according 
to  the  direction  of  the  vector  a.     In  general,  we  intend  to  consider  the 


156  VECTOR  CALCULUS 

vector  dp  as  having  a  purely  arbitrary  direction  unless  the  contrary  is 
stated. 

EXAMPLES 

(1)  Let 

q  =  "  P2. 
Then 

dq  =  -  [p2  +  2dtS-pa  -  p2]  =  -  2dtSpa  =  -  2Spdp. 

Also  since  q  =  T2p  we  have 

dq  =  2TpdTp=-  2Spdp, 
whence 

dTp/Tp  =  Sdp/p,        or        dTp  =  -  SUpdp. 

(2)  From  the  definition  we  have 

d(qr)  =  dq-r  +  g-dr, 
hence 

d(Tp-Up)  =  dTp-Up+  Tp-dUp  =  dp 

and  utilizing  the  result  of  the  preceding  example,  we  have 

dUp/Up  =  Vdplp. 

Also  we  may  write  dUp  =  —  Vdpp-p/T3p  =  pVdpp/T3p 
=  p~lVpdpjTp,  etc.  This  equation  asserts  that  the  dif- 
'  ferential  of  Up  is  the  part  of  the  arbitrary  differential  of  p 
perpendicular  to  Up,  divided  by  the  length  of  p,  that  is, 
it  is  the  differential  angle  of  the  two  directions  of  p  laid  off 
in  the  direction  perpendicular  to  p  in  the  plane  of  p  and 
dp.     In  case  dp  is  along  the  direction  of  p  itself, 

dUp  =  0. 

(3)  We  have  since 

d{pp~lp)  =  dp  =  dp-p~lp  +  pd{p~l)p  +  pp~ldp 

=  2dp  +  pd(p'l)p9 


DIFFERENTIALS  157 

and  thence 

dp  =  —  pd{p~l)p, 

i.p-i  =  -  p-Hpp-1  =  [p-'Spdp  -  p-WpdpWFp 

=  p-'dp-p/rp. 

That  is,  the  differential  of  p~l  is  the  image  of  dp  in  p  divided 
by  the  square  of  Tp. 
Hence 

diVap)-1  =  (Vap)-lVadp-VapjTWap. 

This  vanishes  if  dp  is  parallel  to  a. 

(4)  If  x  =  —  a2/p  then  dir  =  —  a2p~ldppj  T2p,  and  for  two 
different  values  of  dp,  as  dip,  dip,  we  have 

diir/diTT  =  p~ldipld\p-p. 

Therefore  in  the  process  of  "inverting"  or  taking  the 
"electrical  image  "  we  find  that  the  biradial  of  two  dif- 
ferential vectors  is  merely  reflected  in  p.     Interpret  this. 

(5)    T- =  c  is  a  family  of  spheres  with  a  and  —  a  as 

p  —  a 

limit  points.  For  a  differential  dp  confined  to  the  surface 
of  any  sphere  we  have  then 

Sdp[(p  +  a)-1  -  (p  -  a)-1]  =  0. 

A  plane  section  through  a  can  be  written  Syap  =  0,  in 
which  Syadp  =  0  gives  a  differential  confined  to  the  plane. 
Therefore  a  differential  tangent  to  the  line  of  intersection 
of  any  plane  and  any  sphere  will  satisfy  the  equation 

Vdp[VVya«p  +  a)-1  -  (p  -  a)"1)]  -  0. 

But  the  expression  in  the  ()  is  a  tangent  line  to  any  sphere 
which  passes  through  A  and  —  A.  For  the  equation  of 
such  a  sphere  would  be 

p2  -  2Sadp  -  a2  =  0 


158  VECTOR  CALCULUS    

where  5  is  any  vector,  hence  for  any  dp  along  the  sphere, 
S(p  —  VaS)dp  =  0.  But  (p  +  a)-1  —  (p  —  a)-1  is  parallel 
to  a(p2  +  «2)  -  2pSap  and  5(p  -  Va8)[a(p2  +  a2) 
—  2piSap]  =  —  Sap[p2  —  a2  —  2£pa5].  For  points  on  the 
sphere  the  []  vanishes,  hence  the  vector  in  question  is  a 
tangent  line.  Also  Vttt  is  perpendicular  to  it  or  r,  therefore 
the  differential  equation  above  shows  that  the  tangent  dp 
of  the  intersection  of  the  plane  and  the  sphere  of  the 
system  is  perpendicular  to  a  sphere  through  A  and  —A. 
Hence  all  spheres  of  the  set  cut  orthogonally  any  sphere 
through  A  and  —A. 

(6)  The  equation  SU =  e  is  a  familv  of  tores  pro- 
p—a 

duced  by  the  rotation  of  a  system  of  circles  about  their 
radical  axis.     From  this  we  have 

SU(p  +  a)(p-a)  =  -e, 

VU(p  +  a)(p  -  a)  -  V  (1  -  e2)UVap  =  a. 

Differentiating  the  scalar  equation  we  have 

L      P+   OL 

+  TJ(p  +  a)V-^-  >U(p  -  a)l  =  0 

P  —  OL  J 

or 

Sadp[(p  +  a)-1  -  (p  -  a)-1]  =  0. 

Now  in  a  meridian  section  a  is  constant  so  that 

Vdp[(p  +  a)"1  -  (p  -  a)"1]  -  0 

and  dp  is  for  such  section  tangent  to  a  sphere  through 
A  and  —A. 

EXERCISES 
1.  The  potential  due  to  a  mass  m  at  the  distance  Tp  is  m/Tp  in 


DIFFERENTIALS  159 

gravitation  units.     Find  the  differential  of  the  potential  in  any  direc- 
tion, and  determine  in  what  directions  it  is  zero. 

2.  The  magnetic  force  at  the  point  P  due  to  an  infinite  straight 
wire  carrying  a  current  a  is  H  =  —  2h/Vap.  Find  the  differential  of 
this  and  determine  in  what  direction,  if  any,  it  is  zero.  For  Vdpa  =  0, 
dH  =  0;  for  dp  =  dsVa^Vap,  dH  =  -  Hds/TV<rP;  for  dp  =  dsUVap, 
dH  -  V<rUd8./TV<rp. 

3.  The  potential  of  a  small  magnet  a  at  the  origin  on  a  particle  of 
free  magnetism  at  p  is  u  =  Sap/T3p.     Find  the  variation  in  directions 

Up,  UVap,  UaVap. 

4.  The  attraction  of  gravitation  at  a  point  P  per  unit  mass  in  gravita- 
tion units  is 

a  =   -  Up/T*p. 

Find  the  differential  of  <x  in  the  directions  Up  and  F/3p. 

da  =  -  {pHp  -  SpSpdp)/T5P;    parallel    to    p,     -  2/p3; 

perpendicular,  UV@p/Tsp. 

5.  The  force  exerted  upon  a  particle  of  magnetism  at  p  by  an  element 
of  current  a  at  the  origin  is 

H  =  -  V<xPITsp. 

Then  dH  =  {pWadp  -  3VaPSpdp)/T5P;  in  the  direction  of  p,  37a/p3; 
in  the  direction  Vap,  —  VaUVap/T3p. 

6.  The  vector  force  exerted  by  an  infinitesimal  plane  current  at 
the  origin  perpendicular  to  a,  upon  a  magnetic  particle  or  pole  at  p  is 

a  =  (ap2  -  SpSap)/T*P. 

Find  its  variation  in  various  directions. 

2.  Differential  of  Quaternion.  We  may  define  differen- 
tials of  functions  of  quaternions  in  the  same  manner  as 
functions  of  vectors.     Thus  we  have  T2q  —  qKq  so  that 

2TqdTq  =  d(qKq)  =  [(q  +  dtUq)(Kq  +  dtUKq)  -  qKq] 
=  dtlqUKq  +  UqKq] 
=  qKdq  -f-  dqKq 
=  ZSqKdq  =  2SdqKq. 
That  is, 

dTq  =  SdqUKq  =  SdqUq'1  =  TqSdq/q 
or 

dTq/Tq  =  Sdq/q. 


160  VECTOR  CALCULUS 

In  the  same  manner  we  prove  the  other  following  formulae. 

dUq/Uq  =  Vdq/q,         dSq  =  Sdq,         dVq  =  Vdq, 

dKq  =  Kdq,         S(dUq)/Uq  =  0, 

dSUq  =  SUqV(dq/q)  =  -  S(dq/qUVq')TVUq 

=   TVUqdzq, 
dVUq  =  VUKqV(dqlq), 

dTVUq  =  -  SdUqUVq  =  SUqdzq, 

d-q2=  2Sqdq  +  2Sq-  Vdq  +  2Sdq-  Vq, 

d-q~l  =  —  q~ldqq~l, 

d-qaq-1  =  -  2V -qdq^qVaq-1  =  2V-dq(Va)q-\ 

that  is,  if  r  =  gag-1,  then 

dr  =  2V(dqjq>r)  =  -  2V(q-dq-l-r) 

=  2V(Vdq/q)r  -  2q-V 'V{q-ldq-a)q~l 
dUVq=  V'Vdq/Vq-UVq, 
dzq=  S[dqKUVq-q)]. 

We  define  when  7a  =  1 

ax  =  cos  •  irx/2  +  sin  •  7nc/2  •  a  =  catf  •  %tx; 
thus 

d-a*  =  tt/2-o:^1^. 

If  Ta  #  1,  then 

d-ax  =  dz[log  7W*  +  tt/2 -ax+1/Ta\, 

3.  Extremals.  For  a  stationary  value  of  /(p)  in  the 
vicinity  of  a  point  p  we  have  ay(p)  =  0.  If  /(p)  is  to  be 
stationary  and  at  the  same  time  the  terminal  point  of  p 
is  to  remain  on  some  surface,  or  in  general  if  p  is  to  be  subject 

*Tait,  Quaternions,  3d  ed.,  p.  97. 


DIFFERENTIALS  161 

to  certain  conditioning  equations,  we  must  also  have,  if 
there  is  one  equation,  q(p)  =  0,  dq(p)  =  0,  and  if  there  are 
two  equations,  g(p)  =  0  and  h(p)  =  0,  then  also  dg(p)  =  0, 
dh(p)  =  0.  Whether  in  all  these  different  cases  /(p)  attains 
a  maximum  of  numerical  value  or  a  minimum,  or  otherwise, 
we  will  consider  later. 

EXERCISES 

1.  g(p)  =  (p  —  a)2  -f-  «2  =  0,  find  stationary  values  of  Tp  =  f(j>). 
Differentiating  both  expressions, 

Sdp(p  —a)  =  0  =  Sdpp, 

for  all  values  of  dp.  Hence  we  must  have  dp  parallel  to  V  •  tp  where  t  is 
arbitrary,  and  hence  Srp(p  —  a)  =  0,  for  all  values  of  r.  Therefore 
we  must  have  Vp(p  —  a)  =  0,  or  Yap  =  0,  or  p  =  ya.  Substituting 
and  solving  for  y, 

y  =  1  ±  a/Tcc,         p  =  a  ±  aUa.  . 

2.  g(p)  =  (p  —  «)2  +  a2  =  0.     Find  stationary  values  of  &/3p. 
Sdp(p  -  a)  =  0  =  *S/3ap,       whence       dpP.WjS,       £'T,3(p  -  a)  =  0, 

7/3(p  -  a)  =  0. 

p  -  a  =y0,         y  =  a/T(3,         p  =  a  ±  at//3. 

3.  ^(p)  =  (p  —  a)2  -f  a2  =  0,  &G>)  =  *S/3p  =  0,  find  stationary  values 
of  Tp. 

Sdp(P  -  a)  =  0  =  Spdp  =  Spdp,  whence  S-p0(p  —  a)  «  0  -  £pa/S, 
and  since  £p0  =  0,  p  =  yV-fiVafi. 

p  =  V0VaP(l  ±  V[a2  -  S*a0)/TVal3). 

4.  #(p)  =  p2  —  SapSpp  +  a2  =  0.     Find   stationary   values   of   Tp. 

£dPp  =  o  =  £dp(p  -  «S/8p  -  jSflap), 

p  =  x(aS$p  +  /8/Sap)   =  (a£/3p  +  pSap)/(Sa(3  ±  Ta0), 

whence 

Sap  =  TaSpU/3, 


=  SpU(3(Ua  ±  U0)/(SUaU0  T  1). 


Substituting  in  the  first  equation,  we  find  SpUp,  thence  p. 
5.  Sfip  =  c,  >Sap  =  c',  find  stationary  values  of  Tp. 

SdPp  =  Sadp  =  Spdp  =  0,        p  =  xa  +  y$        and 

z£a/3  -f  2//32  =  c,        xa2  +  ySafi  =  c',        whence  x  and  y. 


162  VECTOR  CALCULUS 

6.  Find  stationary  values  of  Sap  when  (p  —  a)2  -f-  a2  =  0. 

Sctdp  =  0  =  Sdp(p  -  a); 
hence 

p  =  ya  =  a  ±  aJ7a 
and 

Sap  =  a2  ±  aTa. 

7.  Find  stationary  values  for  Sap  when  p2  —  SppSyp  +  a2  =  0. 

Sadp  =  0  =  £dP(p  -  )857P  -  ySfip), 
P  =xa+  fiSyp  +  ySfip,        etc. 

8.  Find  stationary  values  of  TV8p  when 

(p  -  a)2  +  a2  =  0. 

9.  Find  stationary  values  of  SaUp  when 

(p  -  a)2  +  a2  =  0. 

10.  Find  stationary  values  of  SaUpSpUp  when 

Syp  +  c  =  0. 

4.  Nabla.  The  rate  of  variation  in  a  given  direction  of 
a  function  of  p  is  found  by  taking  dp  in  the  given  direction. 
Since  df(p)  is  linear  in  dp  it  may  always  be  written  in  the 
form 

where  $  is  a  linear  quaternion,  vector,  or  scalar  function 
of  dp.     In  case  /  is  a  scalar  function,  $  takes  the  form 

—  Sdpv, 

where  v  is  a  function  of  p,  which  is  usually  independent  of 
dp.  In  case  v  is  independent  of  the  direction  of  dp,  we 
call  /  a  continuous,  generally  differentiable,  function. 
Functions  may  be  easily  constructed  for  which  v  varies 
with  the  direction  of  dp.  If  when  v  is  independent  of  dp  we 
take  differentials  in  three  directions  which  are  not  in  the 
same  plane,  we  have 

pS  -  dipd2pd3p  =  V'dipd2p-Sd3pp  +  V '•  d2pd3p  •  Sdipp 

+  V  -  d3pdipSd2pp 
=  —  V 'd1pd2p'd3f '—  Vd2pd3p-dif 

—  V-d3pdip-d2f. 


DIFFERENTIALS  163 

It  is  evident  that  if  we  divide  through  by  Sdipdipdzp,  the 
different  terms  will  be  differential  coefficients.  The  entire 
expression  may  be  looked  upon  as  a  differential  operation 
upon/,  which  we  will  designate  by  V.     Thus  we  have 

v=  V/  = 

_  ( Vdipdip  -  dz  +  V-  d2pdsp  •  di  +  V-  d^pdip  ■  d2)  ,,  , 
S  •  dipdipdzp 
We  may  then  write 

df(P)  =  -  SdpVfip). 

If  the  three  differentials  are  in  three  mutually  rectangular 
directions,  say  i,  j,  k,  then 

V  =  id/dx  +  jd/dy  +  kd/dz. 

It  is  easy  to  find  V/  for  any  scalar  function  which  is  gener- 
ally differentiate  from  the  equation  for  df(p)  above,  that 
is,  df(p)  =  —  SdpVf.    For  instance, 

VSap  =  -  a,         Vp2  =  -  2p,         VTp  =  Up, 
V(Tp)n  =  nTpn-lUp  =  nTpn~2-p,         V  TVap  =  TJVap-a, 
VSaUp  =  -  p-WUpa,         V  •  SapSpp  =  -  pSd$  -  Vap(3, 

V-log  TVap  =  -^~, 
vap 

VT(p  -  a)-'  =  -  U(p  -  a)lT\p-  a), 
VSaUpS(3Up  =  p-WpVap^P, 
Vlog  Tp=  UP/Tp=  -p~\ 

1 


V(ZpA*)  =  -  p~1UVpa  = 


pUVap 


5.  Gradient.  If  we  consider  the  level  surfaces  of  /(p), 
/(p)  =  C,  then  we  have  generally  for  dp  on  such  surface  or 
tangent  to  it  S dp p  =  0  =  df(p)  where  p.  is  the  normal  of  the 


164  VECTOR  CALCULUS 

surface.  Since  Sdp\7f  —  0  and  since  the  two  expressions 
hold  for  all  values  of  dp  in  a  plane 

M  =  *V/, 

or  since  the  tensor  of  p.  is  arbitrary,  we  may  say  V/(p)  is 
the  normal  to  the  level  surface  of  /(p)  at  p.  It  is  called 
the  gradient  of  /(p),  and  by  many  authors,  particularly  in 
books  on  electricity  and  magnetism,  is  written  grad.  p. 

The  gradient  is  sometimes  defined  to  be  only  the  tensor 
of  V/,  and  sometimes  is  taken  as  —  V/.  Care  must  be 
exercised  to  ascertain  the  usage  of  each  author. 

Since  the  rate  of  change  of  /(p)  in  the  direction  a  is 
—  &*V/(p),  it  follows  that  the  rate  is  a  maximum  for  the 
direction  that  coincides  with  UVf,  hence  the  gradient 

V/(p) 

gives  the  maximum  rate  of  change  off(p)  in  direction  and  size. 
That  is,  TVf  is  the  maximum  rate  of  change  of  /(p)  and 
UVfis  the  direction  in  which  the  point  P  must  be  moved  in 
order  that  /(p)  shall  have  its  maximum  rate  of  change. 

6.  Nabla  Products.  The  operator  V  is  sometimes  called 
the  Hamiltonian  and  it  may  be  applied  to  vectors  as  well 
as  to  scalars,  yielding  very  important  expressions.  These 
we  shall  have  occasion  to  study  at  length  farther  on.  It 
will  be  sufficient  here  to  notice  the  effect  of  applying  V  and 
its  combinations  to  various  expressions.  It  is  to  be  ob- 
served that  VQ  may  be  found  from  dq,  by  writing  dq 
=  $-dp,  then  VQ  =  i$i  +  j$j  +  k$k. 

For  examples  we  have 

Vp  =  {Vdipdzp-dzp  +  Vd2pdsp-dip 

+  Vdzpdip  •  d2p)  I '(—  Sdipd2pd3p) 

=  -  3 

since  the  vector  part  of  the  expression  vanishes. 


DIFFERENTIALS  165 

Vp_1  =  —  (Vdipd2p-p~1d3pp~1  +  •••)/(—  Sd1pd2pd3p) 
Since 


-  -  P"2. 


dUp  =  V^  •  Up,        dTp  =  -  SUpdp. 


Hence 


VUp  =  2iV--Up=  -~,        VTp=  Up. 
p  Tp 

Expressions  of  the  form  2F(i,  i,  Q)  are  often  written 
F($ >  r>  Q)>  a  notation  due  to  McAulay. 

Vap  =  a, 

VfaSfap  +  cx2S/32p  +  mSfop)  =  -  0m  +  /52a2  +  1830:3), 

VFap  =  2a,         VVap(3  =  &xft 

VSapVfip  =  -  Sapp  +  3/3£a<p  -  pSa(3, 

VVaUp=  (a  +  p^Sap)/  Tp,         V  •  TTap  =  C/Fap  •  a, 

VTVpVap  -  (Fap  +  ap)UVpVap, 

VVap/T3p  m  (ap2  -  SpSap)/T5p, 

V'UV«P=Tkp>        VUVpVap  =  ^P-, 

V(Vap)-i=0,         V(g)=0. 

EXERCISE 

Show  that  (Fa/3 -<l>y  -+-  y0y<£>a  +  Vy<x'3?P)/Sa0y  is  independent 
of  a,  /3,  7,  where  $  is  any  rational  linear  function  (scalar,  vector,  or 
quaternion)  of  the  vector  following  it.  If  <*>  =  S8( )  +  2ai<S/3i( )  the  ex- 
pression is  5  +  S/Siai. 

Notation  for  Derivatives  of  Vectors 
Directional  derivative 
-  SaV,  Tait,  Joly. 
a- V,  Gibbs,  Wilson,  Jaumann,  Jung. 

Tp  -a,  Burali-Forti,  Marcolongo. 


166  VECTOR  CALCULUS 

Circuital  derivative 
VaV,  Tait,  Joly. 
a  X  V,  Gibbs,  Wilson,  Jaumann,  Jung. 

Projection  of  directional  derivative  on  the  direction. 
S-<rlvSau,  Tait,  Joly. 

—  >  Fischer. 
da 

Projection    of   directional    derivative    perpendicular    to    the 

direction 

V-trhi'SV'a,  Tait,  Joly. 

——  *  Fischer. 
da 

Gradient  of  a  scalar 

V,  Tait,  Joly,  Gibbs,  Wilson,  Jaumann,  Jung,  Carvallo, 

Bucherer. 
grad,  Lorentz,  Gans,  Abraham,  Burali-Forti,  Marcolongo, 

Peano,  Jaumann,  Jung. . 

—  grad,  Jahnke,  Fehr. 

[Fischer's  multiplication  follows  Gibbs,  d/dr 
d    p.    ,  being  after  the  operand,  the  whole  being 

dr  read   from    right   to    left;    e.g.,    Fischer's 

Vfl  is  equiv.  to  —  vSV.] 

Gradient  of  a  vector 

V,  Tait,  Joly,  Gibbs,  Wilson,  Jaumann,  Jung,  Carvallo. 
grad,  Jaumann,  Jung. 

-=-  >  Fischer. 
dr 

7.  Directional  Derivative.     One  of  the  most  important 
operators  in  which  V  occurs  is—  SaV,  which  gives,  the 


DIFFERENTIALS  167 

rate  of  variation  of  a  function  in  the  direction  of  the  unit 
vector  a.     The  operation  is  called  directional  differentiating. 

SaV'Sfo  =  -  SaP,         SaV-p2  =  -  2Sap, 
SaVTp  -  SaUp,  SaVTp-1  =  -  Sap/Tp*  =  UY^p-2, 

SaVTVap=  0,         SaV-Up=  -  ^~  • 

An  iteration  of  this  operator  upon  Tp~l  gives  the  series  of 
rational  spherical  and  solid  harmonics  as  follows : 

-  SaVTp-1  =  -  Sap/Tp*  =  UYiTff*, 
Sl3VSaVTp-1=  (3SapS(3p+  Tp2Sa(3)Tp-5  =  2\Y2Tp~\ 
SyVSWSaVTp-1  =  -  (3.5SapS(3pSyp 

+  32S(3ySapTp2)Tp-7  =  3\Y3Tp~\ 

For  an  n  axial  harmonic  we  apply  n  operators,  giving 

Yn  =  S.(-  l)8(2n  -  2s)!/[2n-*nl(n  -  s)l\ESn-28aUpSsa1a2, 

0  ^  s^  n/2. 

The  summation  runs  over  n  —  2s  factors  of  the  type 
SaiUpSoi2Up •  -  •  and  s  factors  of  the  type  SajCtjSotnar  -  - , 
each  subscript  occurring  but  once  in  a  given  term.  The 
expressions  Y  are  the  surface  harmonics,  and  the  expressions 
arising  from  the  differentiation  are  the  solid  harmonics 
of  negative  order.  When  multiplied  by  Tp2n+1  we  have 
corresponding  solid  harmonics  of  positive  order. 

The  use  of  harmonics  will  be  considered  later. 

8.  Circuital  Derivative.  Another  important  operator  is 
Va\7  called  the  circuital  derivative.  It  gives  the  areal 
density  of  the  circulation,  that  is  to  say,  if  we  integrate 
the  function  combined  with  dp  in  any  linear  way,  around 
an  infinitesimal  loop,  the  limit  of  the  ratio  of  this  to  the 
area  of  the  loop  is  the  circuital  derivative,  a  being  the  normal 
to  the  area.     We  give  a  few  of  its  formulae.     We  may  also 


168  VECTOR  CALCULUS 

find  it  from  the  differential,   for  if  dQ  =  $dp,    Fa  V  •  Q 

VaV  •  Tp  -  VaUp,         FaV  •  Tpn  =  nTpn~2Vapt 

VaV  -  Up  =  (Sap2  -  pSap)/Tp\         VaV-SQp  =  F/3a, 

Fa  V  •  V(3p  =  a(3+  S-aP,         FaV  -ft>  =  2Sa(3, 

FaV  •  7Tft>  -  -  V-apUVpp,         FaV  -p  -  -  2a, 

Fa V  •  (aiSftp  +  a2»S/32p  +  a3S/33p)  -  Sa(« A  +  "A 

+  a3fo)  +  FaiFa/3i  +  Fa2Fa/32  +  Fa3Fa/33. 

9.  Solutions  of  VQ  =  0,  V2Q  =0.  In  a  preceding 
formula  we  saw  that  V(Vap)~l  =  0.  We  can  easily  find 
a  number  of  such  vectors,  for  if  we  apply  Sa  V  to  any  vector 
of  this  kind  we  shall  arrive  at  a  new  vector  of  the  same 
kind.  The  two  operators  V  and  Sa V  •  are  commutative 
in  their  operation.     For  instance  we  have 

d(Vap)~l  =  -  (VapyWadp-iVap)-1; 
hence 

T  =  ^V-(Fap)-1  =  {Vap)-lV$a>{Vap)-1 

is  a  new  vector  which  gives  Vr  =  0.  The  series  can  easily 
be  extended  indefinitely.  Another  series  is  the  one  de- 
rived from  Up/T2p.  This  vector  is  equal  to  p/T3p,  and  its 
differential  is 

(-p2dp+SSdpp.p)/T% 

The  new  vector  for  which  the  gradient  vanishes  is  then 

(-ap2+3Sap-p)/7V 

The  latter  case  however  is  easily  seen  to  arise  from  the 
vector  V  Tp~l,  and  hence  is  the  first  step  in  the  process  of 
using  V  twice,  and  it  is  evident  that  S72Tp~l  =  0.  So  also 
the  first  case  above  is  the  first  step  in  applying  V2  to  log 
TVap-a~l  so  that  V2(log  TVap-a)  =  0.  Functions  of  p 
that  satisfy  this  partial  differential  equation  are  called 


DIFFERENTIALS  169 

harmonic  functions.  That  is,/(p)  is  harmonic  if  V2/(p)  =  0. 
Indeed  if  we  start  with  any  harmonic  scalar  function  of  p 
and  apply  V  we  shall  have  a  vector  whose  gradient  van- 
ishes, and  it  will  be  the  beginning  of  a  series  of  such  vectors 
produced  by  applying  &*iV,  Sa2  V,  •  •  •  to  it.  However  we 
may  also  apply  the  same  operators  to  the  original  harmonic 
function  deriving  a  series  of  harmonics.  From  these  can 
be  produced  a  series  of  vectors  of  the  type  in  question. 
V2  •  F(p)  is  called  the  concentration  of  F(p) .  The  concentra- 
tion vanishes  for  a  harmonic  function. 

EXERCISES 

Show  that  the  following  are  harmonic  functions  of  p: 
1.  Tp-1  tan"1  Sap/Spp, 

where  a  and  /?  are  perpendicular  unit  vectors, 


2. 

Tf*  log  tan  ^  Z  £ 

3. 
where 

and 

tan-1  Sap/S/3p 

Sa(3  =  0 
a2  =  £2  =    _  1# 

4. 

logtan^  z  -  • 

£j            CL 

10.  Harmonics.     We  may  note  that  if  u,  v  are  two  scalar 
functions  of  p,  then 

V  -uv  =  u  Vfl  +  v\7u 
and  thus 

V2-uv  =  u\7h  +  vV2u  +  2SVuVv. 

Hence  the  product  of  two  harmonics  is  not  necessarily 
harmonic,  unless  the  gradient  of  each  is  perpendicular  to 
the  gradient  of  the  other. 
Also  if  u  is  harmonic,  then 

\72-uv  =  u\72v  +  2SVu\7v. 

12 


170  VECTOR  CALCULUS 

If  u  is  harmonic  and  of  degree  n  homogeneously  in  p,  then 
w/7p2n+I  is  a  harmonic  of  degree  —  (n  +  1).     For 

V2(fp2n+1)-1  .    V[_    (2n+    l)prp-2n-3] 

=  -  (2n+  l)(2n)Tp~2n-3 
and 
SVuVTp-2"-1  =  -  (2n+  l)Tp-2n-*SVup 

=  (2n+  l)(2n)uTp-2n-*; 
hence 

V2-u/Tp2n+1  -  0. 

In  this  case  w  is  a  solid  harmonic  of  degree  n  and  uTp~2n~l 
is  a  solid  harmonic  of  degree  —  n  —  1.     Also  uTp"11  is  a 
corresponding  surface  harmonic.     The  converse  is  true. 
EXAMPLES    OF  HARMONICS 

Degree  n  =  0;  <p  =  tan-1  — - 

£>pp 

where  Sc&  =  0,  a2  =  /32  =  -  1; 

^  =  log  cot  ^/  -a2  =  -  1; 
a 

S-a(3UpSapS(3p/V2-a(3p; 

Sa(3UpS(a  +  0)pS(a  -  /3)p/F2a/3p. 

The  gradients  of  these  as  well  as  the  result  of  any  opera- 
tion Sy  V  are  solid  harmonics  of  degree  —  1,  hence  multiply- 
ing the  results  by  Tp[n  =  1,  2n  —  1  =  1]  gives  harmonics 
again  of  degree  0.  These  will  be,  of  course,  rational 
harmonics  but  not  integral. 

Taking  the  gradient  again  or  operating  by  $71 V  any 
number  of  times  will  give  harmonics  of  higher  negative 
degree.  Multiplying  any  one  of  degree  —  n  by  Tp2n~1 
will  give  a  solid  harmonic  of  degree  n  —  1. 

Degree  n  =  —  1.  Any  harmonic  of  degree  0  divided  by 
Tp,  for  example, 

1/Tp,  ip/Tp,         f/Tp,         Saf3UpSaUpS(3p/V2a(3p,  •  •  • , 


DIFFERENTIALS  171 

Degree  n  =  —  2. 

SaUp/p2,         <pSa(3Up/p2,         xPSa(3Up/p2  +  P"2  •  •  • . 
Degree  n  =  1. 

Sop,         *>&*ft>,         ^Softa  +  7p  •  •  • . 

Other  degrees  may  easily  be  found. 

11.  Rational  Integral  Harmonics.  The  most  interesting 
harmonics  from  the  point  of  view  of  application  are  the 
rational  integral  harmonics.  For  a  given  degree  n  there 
are  2n  +  1  independent  rational  integral  harmonics.  If 
these  are  divided  by  Tpn  we  have  the  spherical  harmonics 
of  order  n.  When  these  are  set  equal  to  a  constant  the 
level  surfaces  will  be  cones  and  the  intersections  of  these 
with  a  unit  sphere  give  the  lines  of  level  of  the  spherical 
harmonics  of  the  given  order.  A  list  of  these  follow  for 
certain  orders.  Drawings  are  found  in  Maxwell's  Electricity 
and  Magnetism. 

Rational  integral  harmonics,  Degree  1.  Sap,  S(3p,  Syp, 
a,  ft,  y  a  trirectangular  unit  system. 

Degree  2.     SapS(3p,  SfoSyp,  SypSap,  3S2ap  +  p2,  S2ap 

-  s2pP. 

These  correspond  to  the  operators  7p5[£27V,  SyVSaV, 
SyVSPV,  S(a  +  0) VS(a  -  0) V,  SaVSQV]  on  Tp'K 

Degree  3.  Representing  Sap  by  —  x,  Sfip  by  —  y,  Syp  by 
—  z,  SaV  by  —  Dx,  S(3V  by  —  Dy,  SyV  by  —  Dz  we  have 

2z3  —  3x2z  —  3y2z,        4:Z2x  —  x*  —  y2x,        A.z2y  —  x2y  —  y3, 

x2z  —  y2z,        xyz,         xz  —  3xy2,         3x2y  —  y3 

corresponding  to 

7)3        7)3        7)3        7)        3        _    7)        3         7)        3         _    Q7)        3 

^  zzz  )      -lszzx  ,      Lf  zzy  ,      ^xxz  >  J^xyz  ,      ^xxx  >  OUXyy  , 

7)        3  _    Q7)        3 

■LSyyy  j  OJ^xxy  • 


172  VECTOR  CALCULUS 

Degree  4. 

3z4  +  3y4  +  8z4  +  6*y  -  24z2z2  -  24yV, 

*z(4z2  -  Sx2  -  3y2),        yz(4z2  -  3^  -  3i/2) 

(^  _  y2)(6z2  —  x2  —  y2),        xy(6z2  —  x2  —  y2), 

xz(x>  -  Sy2),        yzQx2  -  y2),        x*  +  y*  -  My2, 

xyix2  -  y2) 


7)           4                7)           4                7)           4 
is zzzz  )             *-* zzzx  y             -L/zzzy 

D      4  - 

■LS  ZZXX 

.    T)         4                 7)          4                 7)          4  _   OT)          4 

*s zzyy  j                M* zzxy  i                J^xxxz            *->±sXyyz  j 

7)       4 

1Jyyyz 

_    Q7)          4                7)          417)          4_  ft/)          4 
oiyxxyz  )           Uxxxx    T  ^  yyyy          ^^xxyy  y 

D      4  —  7)      4 

J^xxxy            ^xyyy  • 

The  curves  of  the  intersections  of  these  cones  with  the 
unit  sphere  are  inside  of  zero-lines  as  follows : 

Degree  1.     Equator,  standard  meridian,  longitude  90°. 

Degree  2.  Latitudes  ±  sin-1  JV  3,  equator  and  standard 
meridian,  equator  and  longitude  90°,  longitude  ±  45°. 
Standard  meridian  and  90°. 

Degree  3.  Latitudes  0°,  db  sin-1  V  0.6,  latitudes  ±  sin-1 
V  0.2  and  standard  meridian,  latitudes  ±  sin-1  V  0.2  and  90° 
longitude,  equator,  longitude  ±  45°,  equator,  longitudes 
0°,  90°.     Longitudes  ±  30°,  90°,  longitudes  ±  60°,  0°. 

12.  Variable  System  of  Trirectangular  Unit  Vectors. 
We  will  consider  next  a  field  which  contains  at  every  point 
a  system  of  three  lines  which  are  mutually  perpendicular. 
That  is,  the  lines  in  one  direction  are  given  by  a,  say,  at  the 
same  point  another  set  by  ft  and  the  third  set  by  y.  Each 
is  a  given  function  of  p,  subject  to  the  conditions 

a/3  =  7,         /?7  =  a,         ya  =  /?,         a2  =  (32  =  y2  =  —  1. 

For  example,  in  the  ordinary  congruence,  a  being  the  unit 
tangent  at  any  point  of  one  line  of  the  congruence,  then 
the  normal  and  the  binormal  are  determined  and  would 
be  ]S  and  7.     However  /?  and  7  may  be  other  perpendicular 


DIFFERENTIALS  173 

lines  in  the  plane  normal  to  a.  If  we  follow  the  vector 
line  for  /3  after  we  leave  the  point  we  shall  get  a  determinate 
curve,  provided  we  consider  a  to  be  its  normal.  We  may 
however  draw  any  surface  through  the  point  which  has 
a  for  its  normal  and  then  on  the  surface  draw  any  curve 
through  the  point.  All  such  curves  can  serve  as  ft  curves 
but  a  might  not  be  their  principal  normal.  It  can  happen 
therefore  that  the  j8  curves  and  the  y  curves  may  start  out 
from  the  point  on  different  surfaces.  However  a,  (3,  and  y 
are  definite  functions  of  the  position  of  the  point  P,  with 
the  condition  that  they  are  unit  vectors  and  mutually 
perpendicular. 

If  we  go  to  a  new  position  infinitesimally  close,  a  becomes 
a  +  da,  ft  becomes  fi  +  dp,  and  y  becomes  y  +  dy.  The 
new  vectors  are  unit  vectors  and  mutually  perpendicular, 
hence  we  have  at  once 

S-ada  =  S-pdp  =  S>ydy  =  0,         Sadp  =  -  S(3da,  n  . 
Spdy  =  -  Sydp,         Syda  =  -  Sady.  {L) 

These  equations  are  used  frequently  in  making  reductions. 
We  have  likewise  since  a2  =  —  1, 

Va-a  =  -  VW,         V/3-/3  =  -  VW,  (2) 

vy-t  =  -  v'rr'j 

where  the  accent  on  the  V  indicates  that  it  operates  only 
on  the  accented  symbols  following.     Similarly  we  have 

Va-j8  +  V(3-a=  -  V'a0'  -  V'j&x',         etc.       (3) 

We  notice  also  that 

S-a(SQV)a  =  0, 
S-a(SQV)0  =  -  S-p(S()V)a,        etc.  (4) 

We  now  operate  on  the  equation  y  =  afi  with  V,  and 


174  VECTOR   CALCULUS 

remember  that  for  any  two  vectors  X/x  we  have  X/x  =  —  juX 
+  2<SX/x,  whence 

V7  =  Va-j3  +  V'aP'  =  Va-/3  -  V/3-a  +  2V'Sa(3'.     (5) 

The  corresponding  equations  for  the  other  two  vectors  are 
found  by  changing  the  letters  cyclically. 
Multiply  every  term  into  y  and  we  have 

Vt-7  =  Vo-a  +  Vj8-|8  +  2V'Sct(3'-y.  (G) 

If  now  we  take  the  scalar  of  both  sides  we  have 

SyVy  =  SaVa  +  Sj8V0  +  2SyV'Sa(3'.  (7) 

We  set  now 

2p  =  +  &*Va  +  SjSVjS  +  #7  Vt  (8) 

and  the  equation  (7)  gives,  with  the  similar  equations 
deduced  by  cyclic  interchange  of  the  letters, 

SyVSctP  -  -  SyV'Sa'Q  =  -  p  +  S7V7, 
SaV'SPy'  m  -  SaV'Sfi'y  =  -  P  +  5a V«, 
SpVSya'  =  -  SpV'Sy'a  =  -  p  +  S0V/3, 
-  S-Tf-  5a  V  •  y]  =  5a V  •  £77'  =  |&*  V  ■  72  =  0,     (  j 
-  5-a[-  5aV-7]  =  -  SaV-Sa'y 

=  Sy(—  SaV  -a)  =  Sy(u(3  +  vy)  =  —  v. 

That  is  to  say,  the  rate  of  change  of  y,  if  the  point  is  moved 
along  a,  is  ]8(5aVa  —  p).     Likewise 

dfi/ds  =  —  7(—  p  +  5aVa)— -ya. 

The   trihedral   therefore   rotates   about   a  with   the    rate 
(p  —  SaVa)  as  its  vertex  moves  along  a.     Now  we  let 

ta  =  +  p  -  SaVa.  (10) 

We  may  also  write  at  once,  similarly, 

h  -  +  V  -  S0VA        *7  =  +  p  -  57V7,       (10) 
from  which  we  derive 

t  +  V+ <»-+■*  (ID 


DIFFERENTIALS  175 

It  is  also  evident  that 

*.  +  U  =  SrVy,        tfi  +  *,  -  SaVa,  /7  +  /a  =  5/3 V/3.   (12) 

The  expressions  on  the  left  hold  good  for  any  two  per- 
pendicular unit  vectors  in  the  plane  normal  to  the  vector 
on  the  right,  and  hence  if  we  divide  each  by  2  and  call  the 
result  the  mean  rotatory  deviation  for  the  trajectories  of  the 
vector  on  the  right,  we  have 

TjSctVct  =  mean  rotatory  deviation  for  a. 

Again  the  negative  rotation  for  the  0  trajectory  gives 
what  we  have  called  previously  the  rotatory  deviation  of  a 
along  j3.  Hence,  as  a  similar  statement  holds  for  y,  the 
mean  rotatory  deviation  is  one  half  the  sum  of  the  rotatory 
deviations.  Hence  %Sa\7a  is  the  negative  rate  of  rotation 
of  the  section  of  a  tube  of  infinitesimal  size,  whose  central 
trajectory  is  a,  about  a,  as  the  point  moves  along  a.  Or 
we  may  go  back  to  (9)  and  see  that 

SaVa  =  (+  p  ~  SPVB)  +  (+  V  ~  SyVy) 

=  -  SpV'Sya'  +  SyV'Sfa', 

which  gives  the  rotatory  deviations  directly. 
The  scalar  of  (5)  and  the  like  equations  are 


SVa  =  SyVP  -  Sj3\7y,    SVP  =  SaVy  -  SyVa, 


(13) 


SVy  =  SfiVa  -  SaVP, 
We  multiply  next  (5)  by  a  and  take  the  scalar,  giving 


SyVa  =  -  SaV'Sfia'  =  SaV'Sa(3f, 
SfiVa  m  -  SaVSay*  =  SaV'Sya', 
SaVP  =  -  SpV'Sy?  =  St3V'S(3y', 
£TV/3  =  -  SpV'Spa'  =  S(3V'Sa(3', 
SfiVy  =  -  SyV'Say'  =  SyV'Sya', 
SaVy  =  -  SyV'Sy(3'  m  SyV'Sfiy'. 


(14) 


176  VECTOR  CALCULUS 

We  can  therefore  write 

SVa  =  -  SWSPa'  -  SyV'Sya', 

that  is  SVa  equals  the  negative  sum  of  the  projection  of 
the  rate  of  change  of  a  along  (3  on  /3,  and  the  rate  of  change 
of  a  along  y  on  y.  But  these  are  the  divergent  deviations 
of  a  and  hence  —  SVa  is  the  geometric  divergence  of  the 
section.  It  gives  the  rate  of  the  expansion  of  the  area  of 
the  cross-section  of  the  tube  around  a.  We  may  write  the 
corresponding  equations  of  /8  and  y. 
Again  we  have 

FVa   =  —  aSaVa  —  (3S(3Va  —  ySyVoc 

=  cx(ta  -  v)  -  PSy(SaV-a)  +  ySp(SaV-a) 
=  a(ta  —  p)  —  Va(SaV-a). 

Now  from  the  Frenet  formulae 

—  Sa'V  'Ol  =  cav, 

where  ca  is  the  curvature  of  the  trajectory  and  v  is  the 
principal  normal.     Hence 

Wa  =  a(ta  -  p)  +  CJh  (15) 

where  /i  is  the  binormal  of  the  trajectory.  We  find  there- 
fore that  VVd  consists  of  the  sum  of  two  vectors  of  which 
one  is  twice  the  rate  of  rotation  of  the  section  or  an  elemen- 
tary cube  about  a,  measured  along  a,  and  the  other  is  twice 
the  rate  of  rotation  of  the  elementary  cube  about  the 
binormal  measured  along  the  binormal.*     But  we  will  see 

*  This  should  not  be  confused  with  the  rotation  of  a  rigid  area  mov- 
ing along  a  curve.  The  infinitesimal  area  changes  its  shape  since  each 
point  of  it  has  the  same  velocity.  As  a  deformable  area  it  rotates  (i.e. 
the  invariant  line  of  the  deformation)  with  half  the  curvature  as  its 
rate.  The  student  should  picture  a  circle  as  becoming  an  ellipse, 
which  ellipse  also  rotates  about  its  center. 


DIFFERENTIALS  177 

later  that  this  sum  is  the  vector  which  represents  twice  the 
rate  of  rotation  of  the  cube  and  the  axis  as  it  moves  along 
the  trajectory  of  a.  Hence  this  is  what  we  have  called 
the  geometric  curl. 

We  may  now  consider  any  vector  a  defining  a  vector 
field  not  usually  a  unit  vector.     Since  a  =  TaUa,  we  have 

SVa  =  SUaVTa  +  TaSvUa. 

The  last  term  is  the  geometric  convergence  multiplied  by 
the  length  of  a,  that  is,  it  is  the  convergence  of  a  section 
at  the  end  of  a.  The  first  term  is  the  negative  rate  of 
change  of  TV  along  a.  The  two  together  give  therefore 
the  rate  of  decrease  of  an  infinitesimal  volume  cut  off  from 
the  vector  tube,  as  it  moves  along  the  tube.  In  the  lan- 
guage of  physics,  this  is  the  convergence  of  a.  Similarly 
we  have 

Wa=  VvTaU<r+  TaWUa. 

The  last  term  is  the  double  rate  of  rotation  of  an  elementary 
cube  at  the  end  of  a,  while  the  first  term  is  a  rotation  about 
that  part  of  the  gradient  of  Ta  which  is  perpendicular  to 
Ua.  It  is,  indeed,  for  a  small  elementary  cube  a  shear  of 
one  of  the  faces  perpendicular  to  Ua,  which  gives,  as  we 
have  seen,  twice  the  rate  of  rotation  corresponding.  Con- 
sequently VVa  is  twice  the  vector  rotation  of  the  elemen- 
tary cube. 

EXAMPLES 

(1)  Show  that 

aSVa  +  (3S  V0  +  yS  Vt 

=  -  VaWa  -  V(3WP  -  VyV\7y. 

(2)  Show  that  if  dipt)  is  zero  VaWa  =  0.  This  is  the 
condition  that  the  lines  of  the  congruence  be  straight.  It 
is  necessary  and  sufficient. 


178  VECTOR  CALCULUS 

(3)  Let  Wot  -  f,  -  SaVa  «  z,  then  Tf  =  V  [c2  +  *% 

fi  =  —  &x  V  •  £  =  #ia  +  c^/3  +  c%yt  where  the  subscript 
1  means  differentiation  as  to  s,  that  is,  along  a  line  of  the 
congruence. 

-  S^  -  cip;        a!  =  cr'Sfei  +  x, 
or 

This  gives  the  torsion  in  terms  of  the  curl  of  a  and  its 
derivative. 

(4)  If  the  curves  of  the  congruence  are  normals  to  a  set 
of  surfaces,  then 

a  =  UVu       and       V«  =  V2u/TVu  -  V(l/TVu)-Vu. 

Hence  we  have  at  once  SaVa  =  0  =  x.     This  condition 
is  necessary  and  sufficient. 

(5)  If  also  VaWoi  —  0,  we  have  a  Kummer  normal 
system  of  straight  rays.  In  this  case  by  adding  the  two 
conditions,  aV\/a  =  0,  that  is,  Wot  =  0.  This  condition 
is  also  necessary  and  sufficient. 

(6)  If  the  curves  are  plane,  «i  =  0  or  Sa\7a  =  $/3V/3 
+  SyVy  or  $/?£i  =  —  xci,  which  is  necessary  and  suffi- 
cient. 

(7)  If  further  they  are  normal  to  a  set  of  surfaces  S8VP 
+  SyVy  =  0  =  jS|8f i.     The  converse  holds. 

(8)  If  Ci  is  constant,  Sy£i  =  0  and  conversely. 

If  also  plane,  and  therefore  circles,  #/3£i  =  0  or  £i  =  X\a 
+  C\x(3.     This  is  necessary  and  sufficient. 
For  a  normal  system  of  circles  we  have  also 

VVa  =  const  =  C\y. 

(9)  For  twisted  curves  of  constant  curvature  £i  =  —  ciaifi. 


differentials  179 

Notations 
Vortex  of  a  vector 
VVu,  Tait,  Joly,  Heaviside,  Foppl,  Ferraris. 

V  X  u,  Gibbs,  Wilson,  Jaumann,  Jung. 

curl  u,  Maxwell,  Jahnke,  Fehr,  Gibbs,  Wilson,  Heaviside, 

Foppl,  Ferraris.     Quirl  also  appears. 
[Vm],  Bucherer. 

rot  u,  Jaumann,  Jung,  Lorentz,  Abraham,  Gans,  Bucherer. 
J  rot  u,  Burali-Forti,  Marcolongo. 

— ; — ,  Fischer. 
dr 

Vort  u,  Voigt. 

(Notations  corresponding  to  VVu  are  also  in  use  by 

some  that  use  curl  or  rot.) 

Divergence  of  a  vector 

—  SVu,  Tait,  Joly.     S\7u  is  the  "convergence"  of  Max- 

well. 

V  •  u,  Gibbs,  Wilson,  Jaumann,  Jung. 

div  u,  Jahnke,  Fehr,  Gibbs,  Wilson,  Jaumann,  Jung, 
Lorentz,  Bucherer,  Gans,  Abraham,  Heaviside,  Foppl, 
Ferraris,  Burali-Forti,  Marcolongo. 

\7u,  Lorentz,  Abraham,  Gans,  Bucherer. 

— ~ ,  Fischer. 
dr 

Derivative  dyad  of  a  vector 

-  SQV-u,  Tait,  Joly. 

•  Vw,  Gibbs,  Wilson. 

•  V ;  u,  Jaumann,  Jung. 

du 

-p=t  Burali-Forti,  Marcolongo. 

aJr 

— ,  Fischer. 
dr 

Du  - ,  Shaw. 


180  VECTOR  CALCULUS 

Conjugate  derivative  dyad  of  a  vector 

—  VS«(),  Tait,  Joly. 
Vm-,  Gibbs,  Wilson. 
V;  u-f  Jaumann,  Jung. 

Ki(),  Burali-Forti,  Marcolongo. 

-j-,  Fischer. 
drc 

Du-,Shaw. 

Planar  derivative  dyad  of  a  vector 
WVuQ,  Tait,  Joly. 
VX(mX  0),  Gibbs,  Wilson. 
V  *u,  Jaumann,  Jung. 

du 
CK      ,  Burali-Forti,  Marcolongo. 

—  x(Du),  Shaw. 

Dispersion.     Concentration 

—  V2,  Tait,  Joly.     V2  is  the  "concentration"  of  Maxwell. 
V2,  Lorentz,  Abraham,  Gans,  Bucherer. 

V-V,  Gibbs,  Wilson,  Jaumann,  Jung. 
div  grad,  Fehr,  Burali-Forti,  Marcolongo. 

—  div  grad,  Jahnke. 

A2,  for  scalar  operands,   1^,      ,*,,    A.  %-        . 
A/,  for  vector  operands,  jBurah-Forti,  Marcolongo. 

-7-5  >  Fischer. 
dr 

Dyad  of  the  gradient.     Gradient  of  the  divergence 

—  VSV,  Tait,  Joly. 
VV-,  Gibbs,  Wilson. 
V;  V,  Jaumann,  Jung. 

grad  div,  Buroli-Forti,  Marcolongo. 


DIFFERENTIALS  181 

Planar  dyad  of  the  gradient.     Vortex  of  the  vortex 

VVVV(),  Tait,  Joly. 

V*V,  Jaumann,  Jung. 

rot2,  Lorentz,  Bucheoer,  Gons,  Abraham. 

curl2,  Heaviside,  Foppl,  Ferraris. 

rot  rot,  Burali-Forti,  Marcolongo. 

13.  Vector  Potential,  Solenoidal  Field.  If  £  =  VVv, 
then  we  say  that  a  is  a  vector  potential  of  £.     Obviously 

£v£  =  SV2<r  =  0. 

The  vector  potential  is  not  unique,  since  to  it  may  be  added 
any  vector  of  vanishing  curl.  When  the  convergence  of  a 
vector  vanishes  for  all  values  of  the  vector  in  a  given  region 
we  call  the  vector  solenoidal.  If  the  curl  vanishes  then 
the  vector  is  lamellar. 

We  have  an  example  of  lamellar  fields  in  the  vector  field 
which  is  determined  by  the  gradient  of  any  scalar  function, 
for  WVu  =  0. 

In  case  the  field  of  a  unit  vector  is  solenoidal  we  see  from 
the  considerations  of  §  12  that  the  first  and  second  divergent 
deviations  of  any  one  of  its  vector  lines  are  opposite.  If 
then  we  draw  a  small  circuit  in  the  normal  plane  of  the 
vector  line  at  P  and  at  the  end  of  dp  a  second  circuit  in 
the  normal  plane  at  p  +  dp,  and  if  we  project  this  second 
circuit  back  upon  the  first  normal  plane,  then  the  second 
will  overlie  the  first  in  such  a  way  that  if  from  P  a  radius 
vector  sweeps  out  this  circuit  then  for  every  position  in 
which  the  radius  vector  must  be  extended  to  reach  the 
second  circuit  there  is  a  corresponding  position  at  right 
angles  to  it  in  which  it  must  be  shortened  by  an  equal 
amount.  It  follows  that  the  limit  of  the  ratio  of  the  areas 
of  the  two  circuits  is  unity.  Hence  if  such  a  vector  tube  is 
followed  throughout  the  field  it  will  have  a  constant  cross- 


182  VECTOR  CALCULUS 

section.  In  the  general  case  it  is  also  clear  that  SVcr  gives 
the  contraction  of  the  area  of  the  tube. 

When  <r  is  not  a  unit  vector  then  we  see  likewise  that 
SVcr  by  §  12  has  a  value  which  is  the  product  of  the  con- 
traction in  area  by  the  TV  -f-  the  contraction  of  TV  multi- 
plied by  the  area  of  the  initial  circuit.  Hence  SVv  repre- 
sents the  volume  contraction  of  the  tube  of  a  for  length  TV 
per  unit  area  of  cross-section.  When  the  field  is  solenoidal 
it  follows  that  if  TV  is  decreasing  the  tubes  are  widening 
and  conversely. 

For  instance,  S\/Up  =  —  2/Tp  signifies  that  per  unit 
length  along  p  the  area  of  a  circuit  which  is  normal  to  p 
is  increasing  in  the  ratio  2/Tp,  that  is,  the  flux  of  Up  is 
increasing  at  the  rate  of  2/  Tp  along  p.  Also  £  •  Vp  =  —  3 
indicates  that  an  infinitesimal  volume  taken  out  of  the 
field  of  p  is  increasing  in  the  ratio  3.  Of  this  the  increase  2 
is  due  to  the  widening  of  the  tubes,  as  just  stated,  the 
increase  1  is  due  to  the  rate  at  which  the  intensity  of  the 
field  is  increasing.  If  the  field  is  a  velocity  field,  the  rate 
of  increase  of  volume  of  an  infinitesimal  mass  is  3  times 
per  second. 

It  is  evident  now  if  we  multiply  SVo"  by  a  differential 
volume  dv  that  we  have  an  expression  for  the  differential 
flux  into  the  volume.  If  a  is  the  velocity  of  a  moving  mass 
of  air,  say  unit  mass,  then  SV<?  is  the  rate  of  compression 
of  this  moving  mass,  and  SVcrdv  is  the  compression  per 
unit  time  of  this  mass,  and  fffSVcrdv  is  the  increase 
in  mass  per  unit  time  of  matter  at  initial  density  or  com- 
pression per  unit  time  of  a  given  finite  mass  which  occupies 
initially  the  moving  volume  furnishing  the  boundary  pf 
the  integral. 

If  r  is  the  specific  momentum  or  velocity  of  unit  volume 
times  the  density,  then  SVr  is  the  condensation  rate  or 


DIFFERENTIALS  183 

rate  of  increase  of  the  density  at  a  given  fixed  point,  and 
SVrdv  is  the  increase  in  mass  in  dv  per  unit  time.     Hence 
SffSVrdv  is  the  increase  in  mass  per  unit  time  in  a 
given  fixed  space. 
Since 

1 

a  —  -t 
c 

where  c  is  density  at  a  point, 

SVo-  =  --SVct  +  -SVt 
cr  c 

e,  „  i  .    B  log  c       d  log  c 

=  _S(TV.logc+_JL_  =  _iL 

=  total  relative  rate  of  change  of  density 

due  to  velocity  and  to  time, 
=  relative  rate  of  change  of  density  at  a 
moving  point. 
SVc-dv=  increase  in  mass  of  a  moving  dv  divided 
by  the  original  density. 
fffSVv-dv  =  increase  in  mass  in  a  moving  volume  per 
unit  of  time  divided  by  original  density, 
=  decrease  in  volume  of  an  original  mass. 

For  an  incompressible  fluid  SVcr  =  0  or  a  is  solenoidal, 
and  for  a  homogeneous  fluid  SVt  =  0  or  t  is  solenoidal. 
In  water  of  differing  salinity  #Vcr  =  0,  SVr  =\=  0.  We 
have  a  case  of  constant  r  in  a  column  of  air.  If  we  take 
a  tube  of  cross-section  1  square  meter  rising  from  the  ocean 
to  the  cirrus  clouds,  we  may  suppose  that  one  ton  of  air 
enters  at  the  bottom,  so  that  one  ton  leaves  at  the  top,  but 
the  volume  at  the  bottom  is  1000  cubic  meters  and  at  the 
top  3000  cubic  meters.  Hence  the  volume  outflow  at  the 
top  is  2000  cubic  meters.     In  the  hydrosphere  a  and  r 


184  VECTOR  CALCULUS 

are  solenoidal,  in  the  atmosphere  r  is  solenoidal.  We 
measure  a  in  m?/sec  and  r  in  tons/ra2  sec.  At  every  sta- 
tionary boundary  <r  and  r  are  tangential,  and  at  a  surface 
of  discontinuity  of  mass,  the  normal  component  of  the 
velocity  must  be  the  same  on  each  side  of  the  surface,  as 
for  example,  in  a  mass  of  moving  mercury  and  water. 

It  is  evident  that  if  a  vector  is  solenoidal,  and  if  we 
know  by  observation  or  otherwise  the  total  divergent  devia- 
tions of  a  vector  of  length  TV,  then  the  sum  of  these  will 
furnish  us  the  negative  rate  of  change  of  TV  along  a. 
Thus,  if  we  can  observe  the  outward  deviations  of  r  in  the 
case  of  an  air  column,  we  can  calculate  the  rate  of  change  of 
TV  vertically.  If  we  can  observe  the  outward  deviations  of 
a  tube  of  water  in  the  ocean  we  can  calculate  the  decrease 
in  forward  velocity. 

EXERCISES. 

1.  An  infinite  cylinder  of  20  cm.  radius  of  insulating  material  of 
permittivity  2  [farad/cm.],  is  uniformly  charged  with  l/207r  electrostatic 
units  per  cubic  cm.  Find  the  value  of  the  intensity  E  inside  the  rod, 
and  also  outside,  its  convergence,  curl,  and  if  there  is  a  potential  for 
the  field,  find  it. 

2.  A  conductor  of  radius  20  cm.  carries  one  absolute  unit  of  current 
per  square  centimeter  of  section.  Find  the  magnetic  intensity  H  inside 
and  outside  the  wire  and  determine  its  convergence,  curl,  and  potential. 

14.  Curl.  We  now  turn  our  attention  to  another  meaning 
of  the  curl  of  a  vector.  We  can  write  the  general  formula 
for  the  curl 

W<t=  -aSUaVUa- pSyVTa  +  y(cT(T+ SfiVTa) 

Let  Ua  =  a'.  These  terms  we  will  interpret,  one  by  one. 
It  was  shown  that  the  first  term  is  a  multiplied  by  the  sum 
of  the  rotational  deviations  of  <r' .  But  if  we  consider  a 
small  rectangle  of  sides  t)dt  =  dip  and  rdu  =  d2p,  then  the 
corresponding  actual  deviations  are 

Sdipd2af        and     —  Sd2pdia' 


DIFFERENTIALS  185 

and  the  sum  becomes 

Sdipdtff'  —  Sd2pdi<r'. 

But  d2a'  is  the  difference  between  the  values  of  a'  at  the 
origin  and  the  end  of  d2p,  and  to  terms  of  first  order  is  the 
difference  of  the  average  values  of  a'  along  the  two  sides 
dip  and  d\p  +  d2p  —  dip.  Likewise  dia  is  the  difference 
between  the  average  values  of  a'  along  the  side  d2p  and  its 
opposite.  Hence  if  we  consider  Sdpa'  for  a  path  consisting 
of  the  perimeter  of  the  rectangle,  the  expression  above  is 
the  value  of  this  Sdpa'  for  the  entire  path,  that  is,  is  the 
circulation  of  <j'  around  the  rectangle.     Hence  the  coefficient 

-  SUaVUa 

is  the  limit  of  the  quotient  of  the  circulation  around  dip  d2p 
divided  by  dtdu  or  the  area  of  the  rectangle. 

If  we  divide  any  finite  area  in  the  normal  plane  of  a  into 
elementary  rectangles,  the  sum  of  the  circulations  of  the 
elements  will  be  the  circulation  around  the  boundary,  and 
we  thus  have  the  integral  theorem 

fSdpa  =  ffSdipd2PV\7<j 

when  Vdipd2p  is  parallel  to  Fy<r.     The  restriction,  we  shall 
see,  may  be  removed  as  the  theorem  is  always  true. 
The  component  of  V\7<r  along  a  is  then 

—  Ua  Lim  j^Sdpcr/area  of  loop 

as  the  area  decreases  and  the  plane  of  the  loop  is  normal  to  a. 
Consider  next  the  term  —  (3SyV  Ta.     It  is  easy  to  reduce 
to  this  form  the  expression 

[-  S*'(SyV)<r  +  Sy(S&'V)<r][-  j8].         >       id  ; 
But  this  is  the  circulation  about  a  small  rectangle  in  the 

13 


ISC.  VECTOR  CALCULUS 

plane  normal  to  /?.  Hence  the  component  of  VVcr  in  the 
direction  0  is 

—  (3  Lim  J'Sdpff/aresi  of  loop  in  plane  normal  to  /?. 

Likewise  the  other  term  reduces  to  a  similar  form  and  the 
component  of  V\7<r  in  the  direction  7  is 

—  7  Lim  tfSdpa/sLYea  of  loop  in  plane  normal  to  7. 

It  follows  if  a  is  any  unit  vector  that  the  component  of 
V\7(T  along  a  is 

—  a  Lim  JfSdpa/sLfesL  of  loop  in  plane  normal  to  a 

as  the  loop  decreases.  The  direction  of  UVS/a  is  then 
that  direction  in  which  the  limit  in  question  is  a  maximum, 
and  in  such  case  TV\7a  is  the  value  of  the  limit  of  the  cir- 
culation divided  by  the  area.  That  is,  TVS/v  is  the  maxi- 
mum circulation  per  square  centimeter. 

Another  interpretation  of  VV<?  is  found  as  follows:  Let 
us  suppose  that  we  have  a  volume  of  given  form  and  that  a 
is  a  velocity  such  that  each  point  of  the  volume  has  an  inde- 
pendent velocity  given  by  a.  Then  the  moving  volume  will 
in  general  change  its  shape.  The  point  which  is  originally 
at  p  will  be  found  at  the  new  point  p  +  cr(p)dt.  A  point 
near  p,  say  p  +  dp,  will  be  found  at  p  +  dp  +  a(p  +  dp)dt, 
and  the  line  originally  from  p  to  p  +  dp  has  become  instead 
of  dp, 

dp  -f-  dt[a(p  +  dp)  —  <r(p)]  =  dp  —  SdpV  'vdt. 

But  this  can  be  written 

dp'  =  dp-  [W-^'dpa'  +  idpSVo-  -  iV(W(r)dp]dt. 

This  means,  however,  that  we  can  find  three  perpendicular 
axes  in  the  volume  in  question  such  that  the  effect  of  the 


DIFFERENTIALS  187 

motion  is  to  move  the  points  of  the  volume  parallel  to  these 
directions  and  to  subject  them  to  the  effect  of  the  term 

dp  +  iV(W(r)dp  dt. 

Now  if  we  consider  an  infinitesimal  rotation  about  the 
vector  e  its  effect  is  given  by  the  form  (du  being  half  of  the 
instantaneous  angle) 

(1  +  edu)p(l  —  edu)  =  p  +  2Vepdu; 

hence  the  vector  joining  p  and  p  +  dp  will  become  the  vector 
joining  p  +  2Vepdu  and  p  +  2Vepdu  +  dp  -f-  2Vedpdu, 
that  is,  dp  becomes  dp  +  2Vedp  du.  We  find  therefore 
that  the  form  above  means  a  rotation  about  the  vector 
UVV<r  of  amount  \TV\7adt,  or  in  other  words  V\/a, 
when  a  is  a  velocity,  gives  in  its  unit  part  the  instanta- 
neous axis  of  rotation  of  any  infinitesimal  volume  moving 
under  this  law  of  velocity,  and  its  tensor  is  twice  the  angu- 
lar velocity.  For  this  reason  the  curl  of  a  is  often  called  the 
rotation.  When  V\/<r  =  0,  a  has  the  form  a  =  \/u,  and  u 
is  called  a  velocity  potential.  If  a  is  not  a  velocity,  we 
still  call  u  a  potential  for  a. 

EXERCISES. 

1.  If  a  mass  of  water  is  rotated  about  a  vertical  axis  at  the  rate  of 
two  revolutions  per  second,  find  the  stationary  velocity.  What  are  the 
convergence  and  the  curl  of  the  velocity?     Is  there  a  velocity  potential? 

2.  If  a  viscous  fluid  is  flowing  over  a  horizontal  plane  from  a  central 
axis  in  such  way  that  the  velocity,  which  is  radial,  varies  as  the  height 
above  the  plane,  study  the  velocity. 

3.  Consider  a  part  of  the  waterspout  problem  on  page  50. 

15.  Vortices.  Since  VVc  is  a  vector  it  has  its  vector 
lines,  and  if  we  start  at  any  given  point  and  trace  the  vector 
line  of  FVo"  such  line  is  called  a  vortex  line.  The  field  of 
FVc  is  called  a  vortex  field.  If  a  vector  is  lamellar  the 
vector  and  the  field  are  sometimes  called  irrotational.     The 


188  VECTOR  CALCULUS 

equation  of  the  vortex  lines  is 

VdpWa  =  0  -  8dp V  a  -  V'Sdpa'  -  -  da  -  V'Sdpa'. 

The  rate  of  change  of  a  then  along  one  of  its  vortex  lines  is 
—  V'Saa'.  Since  SvV^a  —  0,  the  curl  of  a  is  always 
solenoidal,  that  is,  an  elementary  volume  taken  along  the 
vortex  lines  has  no  convergence  but  merely  rotates. 

The  curl  of  the  curl  is  VvVVa  =  VV  —  S/SVa  and 
thus  if  a  is  harmonic  the  curl  of  the  curl  is  the  negative 
gradient  of  the  convergence,  and  if  the  vector  is  solenoidal, 
the  curl  of  the  curl  is  the  concentration  VV. 
EXERCISES 

1.  If  Sa<r  =  0  =  SaV  '<r,  and  if  we  set  <r  =  V-ar,  and  determine  X 
so  that  V-X"  =  t,  then  Xa  is  a  vector  potential  of  the  vector  <r. 

2.  Determine  the  vector  lines  in  the  preceding  problem  for  a.  Also 
show  that  the  derivative  of  X  in  any  direction  perpendicular  to  a  is 
equal  to  the  component  of  a  perpendicular  to  both.     What  is  V2A^? 

3.  If  a  =  wy  and  —  Sy  V  •  w  =  0,  then  either  Xa  or  F/3  will  be 
vector  potentials  of  <r  where  (iy  =  a  and  all  are  unit  vectors  and 
SyV'X  =0  =  SyVY. 

4.  If  the  lines  of  <r  are  circles  whose  planes  are  perpendicular  to  y 
and  centers  are  on  p  =  ty,  and  To  =  f(TVyp),  then  any  vector  parallel 
to  y  whose  tensor  is  F(TVyp),  where  —  f  =  dF/dTVyp  is  a  vector 
potential  of  a.     Is  a  solenoidal? 

5.  If  the  lines  of  <r  are  straight  lines  perpendicular  to  y  and  radiating 
from  p  =  ty  and  T<r.  =  f(TVyp),  then  what  is  the  condition  that  <r  be 
solenoidal?    If  Ta  =  /(tan-1  TVyp/Syp)  a  cannot  be  solenoidal. 

6.  If  a  =/(*Sap,  S0p)-Vyp-y,  then  what  is  FV<r?  Show  that  if/ 
is  a  function  of  tan-1  Sap/Spp,  that  SypVf  is  a  function  of  the  same 
angle,  but  if  /  is  a  function  of  TVyp,  SypV  •/  =  0  and  no  vector  of 
the  form  a  =  f(TVyp)Vyp-y  can  be  a  potential  of  yTVyp.  If 
M  =  Sap/Sfip,  then/0*)  =  -  ./V0*)eW0*2  +  1). 

7.  What  are  the  lines  of  a  =  f(Sap,  Sfip)  Vyp  and  what  is  the  curl? 
If  /  is  a  function  of  TVyp,  so  is  the  curl,  and  if 

F{TVyp)  =  (TVyp)-2fTVyp<pTVypdTVyp 

then  F-TVyp  is  a  vector  potential  of  the  solenoidal  vector  y<pT{Vyp). 
If  /  is  a  function  of  p.  the  curl  is  a  function  of  p.,  and  \f(ji)  Vyp  is  a  vector 
potential  of  7/O*). 

8.  If  <r  is  solenoidal  and  harmonic  the  curl  of  its  curl  is  zero.     If  its 


DIFFERENTIALS  189 

lines  are  plane  and  it  has  the  same  tensor  at  all  points  in  a  line  per- 
pendicular to  the  plane,  then  it  is  perpendicular  to  its  curl. 

9.  The  vector  <r  =  f-  Up,  where  /  is  any  scalar  function  of  p,  is  not 
necessarily  irrotational,  but  SaVv  =  0. 

10.  If  a  vector  is  a  function  of  the  two  scalars  S\p,  Sup  where  X,  p. 
are  any  two  vectors  (constant),  or  if  S\p  =  0,  then  what  is  true  of 

11.  If  S<rV<r  4=  0,  show  that  if  F  is  determined  from  S\7<rVF 
=  —  SaX7 9  then  F  is  the  scalar  potential  of  an  irrotational  vector  r 
which  added  to  <r  gives  a  vector  a',  &cr'V V  =  0.  Is  the  equation  for 
F  always  integrable? 

12.  The  following  are  vectors  whose  lines  form  a  congruence  of 
parallel  rays  f(p)a,  f(Sap)a,  f(Vap)a,  [where/  is  a  scalar  function],  which 
are  respectively  neither  solenoidal  nor  lamellar,  lamellar,  solenoidal. 
The  case  of  both  demands  that  To  =  constant. 

13.  Examples  of  vectors  of  constant  intensity  but  varying  direction 
are 

o-  =  aUp,         aVocp  +«V(62  -  a2V2ap). 

Determine  whether  these  are  solenoidal  and  lamellar. 

14.  If  the  lines  of  a  lamellar  vector  of  constant  tensor  are  parallel 
rays,  it  is  solenoidal.  If  the  lines  of  a  solenoidal  vector  are  parallel 
straight  lines,  it  is  lamellar. 

15.  An  example  of  vectors  whose  convergences  and  curls  are  equal 
at  all  points,  and  whose  tensors  are  equal  at  all  points  of  a  surface,  are 

a(x  +  2yz)  +  &(y  +  Szx)  +  xyy,     and    2yza  +  Szx/3  -f-  y(xy  +  2z) 
and  the  surface  is 

x2  +  y2  -  z2  +  6xyz  =  0. 

Therefore  vectors  are  not  fully  determined  when  their  convergences  and 
curls  are  given.  What  additional  information  is  necessary  to  determine 
an  analytic  vector  which  does  not  vanish  at  oo .'  Determine  a  vector 
which  is  everywhere  solenoidal  and  lamellar  and  whose  tensor  is  12 
for  Tp  m  oo . 

16.  Show  that 

—  eV2<Z  =  limr=0  [average  value  of  q  over  a  sphere  of  radius  r,  less  the 

value  at  the  center]  divided  by  r2. 

—  \V2q  =  average  of  (-  SaV)2q  in  all  directions  a. 

—  xVV2g  =  limr=o  [excess   of  average  value  of  q  throughout  a  small 

sphere  over  the  value  at  the  center}  divided  by  r2. 

17.  Show  by  expansion  that 

a(p  +  8p)  =  a(p)  -  S8pX7  -(r(p) 

-  VSP[-  Sa8p  +  ±S8pVPSa8p]  -  W8pVSJ p* 
=  VVSP[~  |F5pa  +  iSSpVpVSpa]  -  ±8pSV P<r. 


190  VECTOR  CALCULUS 

i 

The  first  expansion  expresses  <r  in  the  vicinity  of  p  in  terms  of  a  gradient 
of  a  scalar  and  an  infinitesimal  rotation.  The  second  expresses  a  in 
the  form  of  a  curl  and  a  translation. 

18.  Show  that  for  any  vector  <r  we  have 

£V(W'V"&r\r"<r/7V)   =0, 

where  the  accents  show  on  what  the  V  acts,  and  are  removed  after  the 
operation  of  the  accented  nabla.  The  unaccented  V  acts  on  what  is 
left.     (Picard,  Traits,  Vol.  I,  p.  136.) 

19.  If  a,  <r2  are  two  functions  of  p,  and  d<n  =  <pi(dp),da2  =  widp), 
show  that 

&<riV  -SaiV   —  S<r2V  -SaiV   =  S(<pi<r2  —  <p2<Ti)^7 . 

16.  Exact  Differentials.  If  the  expression  Sadp  is  the 
differential  of  a  function  u(p),  then  it  is  necessary  that 
Sadp  =  —  SdpVu,  for  every  value  of  dp,  which  gives 

a  =  —  Vw. 

When  a  is  the  gradient  of  a  scalar  function  of  u(p),  u  is 
sometimes  called  a  force-function.  It  is  evident  at  once 
that 

VS7<r  =  0,         or         £FOV)cr  =  0  for  every  v. 

This  is  obviously  a  necessary  condition  that  Sadp  be  an 
exact  differential,  that  is,  be  the  differential  of  the  same 
expression,  u,  for  every  dp.  It  is  also  sufficient,  for  if 
VVa  =  0,  it  will,  be  shown  below  that  a  =  Vu,  and 
SVudp  =  —  du. 

In  general  if  Q(p)  is  a  linear  rational  function  of  p, 
scalar  or  vector  or  quaternion,  then  to  be  exact,  Q(dp)  must 
take  the  form 

Q(dp)  —  —  SdpV  -R(p)  for  every  dp. 
Hence  formally  we  must  have  the  identity 

C()=  -S()V-R(p). 
But  if  we  fill  the  (  )  with  the  vector  form  VvV ,  we  have 
Q(Vi>S7)  =  0  for  every  v. 


DIFFERENTIALS  191 

This  may  be  written  in  the  form 

Q'VV'l  ) .-  0  identically. 

EXERCISES 

1.  Vadp  is  exact  only  when  a  =  a  a  constant  vector.  For  VaV\7  v  =  0 
for  every  v,  that  is  S\(vSS7p-  —  VSav)  =  0  for  every  X,  v,  and  for  X 
perpendicular  to  v  therefore  SXS/Sav  =  0,  or  Sdav  =  0  for  every  v 
perpendicular  to  the  dp  that  produces  da.     Again  if  X  =  v, 

SV*  +  SvVSav  =  0, 

for  every  v.  Therefore  S\/a  =  0  and  Sv\7 Sav  =  0,  or  Sdav  =  0  for 
every  dp  in  the  direction  of  v.  Hence  da  =  0  for  every  dp  and  a  =  a 
a  constant. 

2.  Examine  the  expressions 

S^,  V(Vap)dp,  F.&. 

Integrating  Factor 
If  an  expression  becomes  ezactf  &?/   multiplication   by   a 
scalar  function  of  p,  let  the  multiplier  be  m.     Then 

mQ(W)  =  0, 
where  V  operates  on  m  and  Q,  or 

QWm()  +  mQVV()  =  0, 

where  V  operates  on  m  only  in  the  first  term  and  on  Q 
only  in  the  second.     This  gives  for  Sadp 

SaVmi  )  +  mS(  )Vo-  =  0,       or       VaVm  +  mVV<r  =  0. 

This  condition  is  equivalent  however  to  the  condition 

Sa\7<7  =  0. 

Conversely,  when  this  condition  holds,  we  must  have 

VVa-  =  V(tt, 

where  r  is  arbitrary,  hence  StVv  =  0,  and  Sa\7r  =  0. 
But  r  is  any  variable  vector  conditioned  only  by  being 


192  VECTOR  CALCULUS 

perpendicular  to  FV<r,  hence  we  must  have  for  all  such 
VVt  —  0,  or  a  =  0.  The  latter  is  obviously  out  of  the 
question  and  hence  VVt  =  0,  that  is  t  =  Vw,  or  we  may 
choose  to  write  it  r  =  Vu/u. 

Hence,  VV<r+  VVua/u  =  0  =  Vv(ua),  and  S(ua)dp=0 
is  thus  proved  to  be  exact. 

We  may  also  proceed  thus.  Since  every  vector  line  is 
the  intersection  of  two  surfaces,  say  u  =  0  =  v,  then  we  can 
write  the  curl  of  a,  which  is  a  vector,  in  the  form 

VV<r  =  hVVu\7v, 

and  if  S<tS7<t  =  0,  it  follows  that  we  must  have  a  in  the 
plane  of  Vw,  Vfl  and 

a  =  xVu  -f-  yVv.        Sadp  =  —  xdu  —  ydv. 

But  also 

VVcr  =  VVxVu  +  VVyVv  =  hVVuVv. 

Hence 

SVuVyVv  =  0  =  SVvVxVu. 

These  are  the  Jacobians  of  u,  v,  x  and  u,  v,  y  however,  and 
since  their  vanishing  is  the  condition  of  functional  de- 
pendence, it  follows  that  x  and  y  are  expressible  as  functions 
of  u  and  v.     Hence  we  have 

x(u,  v)du  +  y(u,  v)dv  —  0. 

It  is  known,  however,  that  this  equation  in  two  variables  is 
always  integrable  by  using  a  multiplier,  say  g.  Therefore 
S(ga)dp  =  0  is  exact  for  a  properly  chosen  g.  Further  we 
see  that  ga  =  —  Vw,  or  that  when  SaV.a  =  0,  a  =  mVw. 
If  SVo-  =  0  for  all  points,   then  we  find  easily  that 

a  =  Wr. 
For 

a  =  hVVu\/v, 


DIFFERENTIALS  193 

so  that 

SV<t  =  SvhVuVv  =  0 
and 

h  =  h(u,  v). 

Integrate  h  partially  as  to  u,  giving 

w  =  fhdu  +  f(v), 
then 

Vw  =  hVu  +  fvVv,         VX/wVv  =  hVX/uX/v  =  o\ 

Set  r  =  wX/v  or  —  v X7w  and  we  have  at  once  a  =  VX/j. 
It  is  clear  that  if  we  draw  two  successive  surfaces  W\ 
and  w2  and  two  successive  surfaces  Vi  and  v2,  since 

m„         Aw  ,         „,  Av 

T\/w  =  and         T\7v  = 


Ani  An2 

and  the  sides  of  the  parallelogram  which  is  the  section  of 
the  tube  are  A<?2  =  Arii  esc  6,  Asi  =  An2  esc  6,  and 
area  =  AniAn2  esc  6,  then  TVx  area  =  AwAv,  and  these 
numbers  are  constant  for  the  successive  surfaces,  hence  the 
four  surfaces  form  a  tube  whose  cross-section  at  every  point 
is  inversely  as  the  intensity  of  a.  For  this  reason  a  is  said 
to  be  solenoidal  or  tubular. 

If  Vx/a  =  0  for  all  points  then  we  must  have  a  =  V«. 
For  SvVo-  =  0  and  a  =  gVv,  VX/a  =  VX/gX/v,  hence  g 
is  a  function  of  v,  and  we  may  write 

a  =  X7u. 

If  X7d  =  0,  we  must  have,  since  Sx/c  =  0,  a  =  VX/r,  and 
since  VX7(T  =  0,  a  =  X7u,  whence  X72u  —  0.  Therefore,  if 
X7(T  =  0,  <t  is  the  gradient  of  a  harmonic  function  and  also  the 
curl  of  a  vector  r,  the  curl  of  the  curl  of  r  vanishing.  Also 
if  VX7VX/t  =  0,  since  we  must  then  have  Vx/t  =  X/v,  and 
therefore  SV^Vr  =  0  =  V2fl,  we  can  say  that  if  the  curl 


194  VECTOR  CALCULUS 

of  the  curl  of  a  vector  vanishes  it  must  be  such  that  its  curl 
is  the  gradient  of  a  harmonic  function.     Also  SdpVr=  —dv. 
Functions  related  in  the  manner  of  v  and  r  are  very  im- 
portant. 
Since  in  any  case  SvVVcr  —  0,  we  must  have 

Vv<r  =  VVuVv        or         VV(<r  —  u\/w)  =  0, 

whence 

a  —  uVw  =  Vp, 

so  that  in  any  case  we  may  break  up  a  vector  a  into  the  form 

a  =  Vp  +  uVw. 

It  follows  that  SaV<r  =  SVp\7u\/w.     If  we  choose  u,  w 
and  x  as  independent  variables,  we  have 

Vp  =  PxVx  +  puVu  +  pw  Vw, 
whence 

S<tX7(t  =  pxSVxVuVw, 

and  we  can  find  p  from  the  integral 

p  =  fSaVv/SVxVuVw-dx. 

In  case  SaVcr  =  0,  p  =  constant,  and  a  =  uVw. 

A  theorem  due  to  Clebsch  is  useful,  namely  that  a  can 
always  be  put  into  the  form 

<r  =  Vp  +  VVt,       where       V\/Vp  =  0,    SVFVr  =  0, 

that  is,  <r  can  always  be  considered  to  be  due  to  the  super- 
position of  a  solenoidal  field  upon  a  lamellar  field.  We 
merely  have  to  choose  p  as  a  solution  of 

V2p  =  SVcr, 

for   we   have   at  once   Sv(<r  —  Vp)  =  0,   and   therefore 

o-  —  Vp  =  VVt. 


DIFFERENTIALS  195 

This  may  easily  be  seen  to  give  us  the  right  to  set 

<r  =  Vp  +  (Vv)nr. 

EXAMPLES.     SOLUTIONS   OF   CERTAIN   DIFFERENTIAL 
FORMS 

(1).  SV<t  =  0,  then  a  =  VVr,  and  if  Vv<r  =  0,  <r  =  Vp. 
If  V<r  =  0,  <7  =  Vh  where  V2^  =  0. 

(2).  If  <p  is  a  linear  function  dependent  upon  p  continu- 
ously, and  <pV  =  0,  <p  =  OVvQ-    If  <poV  =  0, 

<Po  =  VV(60VV0), 

8,  do  are  linear  functions.     For  the  notation  see  next  chapter. 

(3).  VVvQ  =  0,  <p  =  -  VSaQ.  If  e(Fv  </>())  =  0, 
<P  =  fcFVO  ~  V-SerO.  If  (FV^())o  =  <W  =  p()  -  V^(). 
Fv^o  =  0,  <?o  =  -  S()V-  Vp. 

(4).  A  particular  solution  of  certain  forms  is  given,  as 
follows : 

*SVo"  =  a,  cr  =  Jap,         Fv<r  =  eat,  a  =  \Vap, 
Vp  =  oc,  p  =  —  Sap,         yXJ  =  ol,   (p  =  —  Sap'Q, 
VV<pQ  =  6,         <p  =  -  iVpdQ,         €(VV<pQ)  -  a, 

?  =  -  &*p.(),      (Fvrio  =  0O,       ip ;-  -  i^oO, 

Fw  =  p{),         <p  -  -  fo7p()  -  V&r(). 

EXERCISES 

1.  Consider  the  cases  o-  =  t  -\-jf(g(p))  +  cfc,  where/  and  gr  have  the 
following  values:  f  =  g,  g2,  g3,  <g,fg,  g~\  g~2,  e«,  log  g,  sin  g,  tan  #,  and 
g  has  the  values  y/r,  (y  -  'ax) /(ay  +  »),  (bx  +  jf)/(a;  -  &y),  x/y, 
—  x/y,  —  y/x,  etc.,  V  (x2  +  y2)  —  a. 

2.  Consider  the  vector  lines  of 

a  =  i  cos  (3-n-r)  +  j  sin  (3xr),        r  =  V  (x2  +  y2). 

3.  Consider  the  significance  of  S-Ua\/Ua  =  0;   give  examples. 

4.  If  rf<r  =  Vt  dp  find  F V  <r.  Likewise  if  da  =  adpd,  da  =  aSpdp,  da 
=  —p2dp,  da  =  Vradp  where  t  is  a  function  of  p. 

17.  Groups.     If  Si,  Sj,  •  •  • ,  Sn  are  any  functions  linear 


196  VECTOR  CALCULUS 

in  V  but  of  any  degrees  in  p,  then  they  form  a  transforma- 
tion group  (Lie's)  if  and  only  if  for  any  two  Si,  S;, 

where  0  is  a  linear  function  of  Si,  S2,  •  •  •  Sn,  and  a,  /? 
arbitrary  vectors.  For  instance,  we  have  a  group  in  the 
six  formal  coefficients  of  the  two  vector  operators 


Si  =  -  V  -  pSpV,       S2  =  -  FpV, 


for 


SaZiSpEi  -  S0Ei&*Ei  =  Sa(3Z2, 
SaE2S/3E2  -  £/3E2&*E2  =  -  &x/3S2, 
&*SiS/3S2  -  S/SSt&xSi  =  -  SapBi. 

The  general  condition  may  be  written  without  a,  /3 : 

Kt  S  E/  -  Si'SZj  -  v  e  0, 

where  the  accented  vector  is  operated  on  by  the  unaccented 
one. 

Integration 
18.  Definition.     We  define  the  line  integral  of  a  function 
of  p,f(p),  by  the  expression 

flf(p)<p{dp)  =  Lim  2f(Pi)(p(dpi),    %  -  1,  •  •  •,  »j 

n  =  00 

where  the  vectors  pi  for  the  n  values  of  i  are  drawn  from  the 
origin  to  n  points  chosen  along  the  line  from  A  to  B  along 
which  the  integration  is  to  take  place,  <p(cr)  is  a  function 
which  is  homogeneous  in  a  and  of  first  degree,  rational  or 
irrational,  dpi  =  pt-  —  pz_i,  and  the  limit  must  exist  and  be 
the  same  value  for  any  method  of  successive  subdivision 
of  the  line  which  does  not  leave  any  interval  finite.  Like- 
wise we  define  a  definite  integral  over  an  area  by  the  expres- 
sion 

ffi(p)<P2{dip,d2p)  =  Lim  2f(j>i)<to(dipit  d2pi), 


INTEGRALS  197 

where  <p2  is  a  homogeneous  function  of  dipi  and  d2p{,  two 
differentials  on  the  surface  at  the  point  pi,  and  of  second 
degree.  A  definite  integral  throughout  a  volume  is  simi- 
larly defined  by 

J%J%.ff(p)<P3(dip,  d2p,  dzp)  =  Lim  2/(p»)¥>g(dipt,  d2pi,  d3pi). 

For  instance,  if  we  consider  /(p)  =  a,  we  have  for  ffadp 
along  the  straight  line  p  =  fi  +  #7,  dp  =  cfo-7  and 

Lim  "Zadx-y    from     #  =  #0  to  x  =  Xi  is  0:7(21  —  Xo), 
hence 

^P  =  «(Pi  -  Po). 

The  same  function  along  the  ellipse  p  =  /3  cos  0  +  7  sin  0, 
where  dp  =  (—  /?  sin  6  +  7  cos  0)d0  has  the  limit 

(a/3  cos  6  -\-  ay  sin  0) 

between  0  =  0O,  6  =  0i,  that  is,  again  a(pi  —  p0). 

EXAMPLES 
(1).  j£«  £dp/p  =  log  TWpo,  for  any  path. 
(2)-  Su  ~  q~ldqq~l  =  qr1  —  g0_1,  for  any  path. 
(3).  The  magnetic  force  at  the  origin  due  to  an  infinite 
straight  current  of  direction  a  and  intensity  /  amperes  is 
H  =  0.2-I-Va/p,  where  p  is  the  vector  perpendicular  from 
the  origin  to  the  line.     In  case  then  we  have  a  ribbon  whose 
right  cross-section  by  a  plane  through  the  origin  is  any 
curve,   we  have  the  magnetic  force  due  to  the  ribbon, 
expressible  as  a  definite  integral, 

H  =  0.2IfVaTdp/p. 

For  instance,  for  a  segment  of  a  straight  line  p  =  a(3  -\-  xy, 
/3,  7  unit  vectors  Tdp  =  dx, 

H  =  0.27 / '(ay  -  xt3)dx/(a2  +  x2) 
=  -  0.2/0 -log  (a2  +  *22)/(a2  +  *i2) 

-f-  0.2 -I-yitsoT1  x2/a  —  tan-1  xj/a), 
=  0.27/3  -log  OA/OB  +  O.27J.  L  AOB. 


198  VECTOR  CALCULUS 

(4).  Apply  the  preceding  to  the  case  of  a  skin  current  in  a 
rectangular  conductor  of  long  enough  length  to  be  prac- 
tically infinite,  for  inside  points,  and  for  outside  points. 

(5).  Let  the  cross-section  in  (4)  be  a  circle 

p  —  b3  —  a(3  cos  6  —  ay  sin  0. 

Study  the  particular  case  when  b  =  0  and  the  origin  is  the 
center. 

(6).  The  area  of  a  plane  curve  when  the  origin  is  in  the 
plane  is 

\TfVpdp. 

If  the  curve  is  not  closed  this  is  the  area  of  the  sector 
made  by  drawing  vectors  to  the  ends  of  the  curve.  If  we 
calculate  the  same  integral  \fVpdp  for  a  curve  not  in  the 
plane,  or  for  an  origin  not  in  the  plane  of  a  curve  we  will 
call  the  result  the  areal  axis  of  the  path,  or  circuit.  This 
term  is  due  to  Koenigs  (Jour,  de  Math.,  (4)  5  (1889),  323). 
The  projection  of  this  vector  on  the  normal  to  any  plane, 
gives  the  projection  of  the  circuit  on  the  plane. 

(7).  If  a  cone  is  immersed  in  a  uniform  pressure  field 
(hydrostatic)  then  the  resultant  pressure  upon  its  surface  is 
"~  2^Vpdp-P,  where  p  is  taken  around  the  directrix  curve. 

(8).  According  to  the  Newtonian  law  show  that  the  at- 
traction of  a  straight  segment  from  A  to  B  on  a  unit  point  at 
0  is  in  the  direction  of  the  bisector  of  the  angle  AOB, 
and  its  intensity  is  2/x  sin  ^AOB/c,  where  c  is  the  perpen- 
dicular from  0  to  the  line. 

(9).  From  the  preceding  results  find  the  attraction  of  an 
infinite  straight  wire,  thence  of  an  infinite  ribbon,  and  an 
infinite  prism. 

(10).  Find  the  attraction  of  a  cylinder,  thence  of  a  solid 
cylinder. 

19.  Integration  by  Parts.     We  may  integrate  by  parts 


INTEGRALS  199 

just  as  in  ordinary  problems  of  calculus.     For  example, 

fysV-adpSpP  =  iVa(B8P8  -  ySfa)  +  \VaVPf*V  pdp, 

which  is  found  by  integrating  by  parts  and  then  adding  to 
both  sides  J*yV  -adpSpp.  The  integral  is  thus  reduced  to 
an  areal  integral.  In  case  y  and  5  are  equal,  we  have  an 
integral  around  a  loop,  indicated  by  J?. 

EXAMPLES 

(1).  SfdpVcxp  =  HdVaS  -  yVay)  -  \Vaf*Vpdp 

+  iSafjVpdp. 

(2).  fy*V.VadpV(3p  =  ilaSPSy'Vpdp  +  pS-afy'Vpdp 

-  5 Sap 5  +  y Softy]. 

(3).  fy*S'VadpV(3p  =  i(Sa8S(35  -  Say  Spy  -  82SaP 

-y2Sap-  S-a(3fysVpdp). 

(4).  JfV-adpVPp  =  U*SpfysVpdp  +  pSafjVpdp 

-  dSa(38  +  y  Softy  +  Sa5S(38 

-  Say  Spy  -  82SaP  +  y2SaP 

-  SaPffVpdp). 

(5).  fysSapSpdp  =  USadSpd  -  SaySPy 

-S-Voftf'Vpdpl 

(6).  ffdpSap  =  itfSad  -  ySay  +  V-affVpdp]. 

(7).  fysVaPSpdp  =  HVadSpb  -  VaySPy  -  SoftffVpdp 

+  PSaJfVpdp]. 

(8).  fysVap-dp  =  i[Va6'B  -  Vayy  +  afy8Vpdp 

+  SaffVpdp]. 

(9).  fjapdp  =  h[a(82  -  y2)  +  2af*Vpdp]. 

As  an  example  of  this  formula  take  the  scalar,  and  notice 
that  the  magnetic  induction  around  a  wire  carrying  a 


200  VECTOR  CALCULUS 

current  of  value  Ta  amperes,  for  a  circular  path  a 

B  -  -  2p.Vap/a2. 
Therefore 

-  fO^Sapdp/a2  =  -  SfdpB  =  -  OSfia^SafVpdp 
=  ATafia~2wr2. 

For  fj,  =  1,  r  =  a,  this  is  OAwC.     This  gives  the  induction 
in  gausses  per  turn. 

(10).  SfSdpw  -  i[S8cp8  -  Sy<py]  +  SeffVpdp. 
(11).  /^prfp  =  h[Vy<py  -  V8<p8  +  <p'f*Vpdp 

+  rmffVpdp]* 
(12).  XVprfp  =  }[**.«  -  ^y.7  +  SeffVpdp 

-  tp'f'Vpdp]  -  m.ffVpdp. 
For  any  lineolinear  form 

SfQip,  dp)  =  hm, «)  -  Q(y,  y)] 

+  ifAQiP,  dp)  -  Q(dP,  p)} 
=  ««(*,  *)  -  Q(r  t)]  +  WSfVpdp. 

(13).  State  the  results  for  preceding  12  problems  for  in- 
tegration around  a  loop. 

(14).  Consider  forms  of  second  degree  in  p,  third  degree, 
etc. 

20.  Stokes*  Theorem.  We  refer  now  to  problem 
page  189,  where  we  have  the  value  of  cro,  a  function  of  po, 
stated  for  the  points  in  the  vicinity  of  a  given  fixed  point. 
If  we  write  <tq  for  the  value  of  a  at  a  given  origin  0,  its 
value  at  a  point  whose  vector  is  dp  is 

o-  =  V5p[-  S<r08p  +  %S8pVS(ro8p]  -  £F5pFVo% 

where  V  refers  only  to  <r0,  and  gives  a  value  of  the  curl  at 

*  wii(v)  =  —  Si(pi  —  Sj<pj  —  Sk<pk.     For  notation  see  Chap.  IX. 


INTEGRALS  201 

the  origin  0.  If  we  multiply  by  ddp  and  take  the  scalar, 
we  have 

Sadbp  =  dSp[Sa08p  -  iS8pVSa0dp]  +  iSSpd6pVV<r0. 

Therefore  if  we  integrate  this  along  the  curve  whose  vector 
radius  is  dp  we  have 

ffcSed&p  =  [So-08p2  -  Saodpi  -  §S6p2VS<ro8p2 

+  iSSpiV Saotpi]  +  %SW<T0fVdpd8p. 

The  last  expression,  however,  is  the  value  of 

$[FVovareal  axis  of  the  sector  between  dpi  and  5p2]. 

Therefore  for  an  infinitesimal  circuit  we  have 

fSvodbp  =  £[FVovareal  axis  of  circuit]  =  SUvVVvo-dA. 

FWo  is  the  curl  of  a  at  some  point  inside  the  loop.  If  now 
we  combine  several  circuits  which  we  obtain  by  subdividing 
any  area,  we  have  for  the  sum  of  the  line  integrals  on  the 
left  the  line  integral  over  the  boundary  curve  of  the  area 
in  question,  and  for  the  expression  on  the  right  the  sum  of 
the  different  values  of  the  scalar  of  the  curl  of  a  multiplied 
into  the  unit  normals  of  the  areas  and  the  areas  themselves 
or  the  area  integral  ffSV\/(rdipd2p.  That  is,  we  have 
for  any  finite  loop,  plane  or  twisted,  the  formula 

fSadp  =  ffSVV(TVdlPd2p. 

This  is  called  Stokes'  Theorem.  It  is  assumed  in  the  proof 
above  that  there  are  no  discontinuities  of  a  or  V\/a, 
although  certain  kinds  of  discontinuities  can  be  present. 
The  diaphragm  which  constitutes  the  area  bounded  by 
the  loop  is  obviously  arbitrary,  if  it  is  not  deformed  over 
a  singularity  of  a  or  V\7a. 

It  follows  that  fSadp  along  a  given  path  is  independent 
of  the  path  when  the  expression  on  the  right  vanishes  for 
X4 


202  VECTOR  CALCULUS 

the  possible  loops,  that  is,  is  zero  independently  of  dip, 
dip,  or  that  is,  V\7<r  =  0.  This  condition  is  necessary  and 
sufficient. 

It  follows  also  that  the  surface  integral  of  the  curl  of  a 
vector  over  a  diaphragm  of  any  kind  is  equal  to  the  circula- 
tion of  the  vector  around  the  boundary  of  the  diaphragm. 
That  is,  the  flux  of  the  curl  is  the  circuitation  around  the 
boundary. 

We  may  generalize  the  theorem  as  follows,  the  expression 
on  the  right  can  be  written  ffSUvVVo-  dA,  where  v  is 
the  normal  of  the  surface  of  the  diaphragm  and  dA  is  the 
area  element.  If  now  we  construct  a  sum  of  any  number 
of  constant  vectors  au  a2,  •  •  •  an  each  multiplied  by  a 
function  of  the  form  Saidp,  Scr^dp,  •  •  •  Scrndp,  we  will  have 
a  general  rational  linear  vector  function  of  dp,  say  <pdp, 
and  arrive  at  the  integral  formula 

fvdp  =  ff<p(VUpV)dA, 

where  the  V  refers  now  to  the  functions  of  p  implied  in  <p. 
This  is  the  vector  generalized  form  of  Stokes'  theorem. 

If  the  surface  is  plane,  Uv  is  a  constant,  say  a,  so  that 
for  plane  paths 

fipdp  =  ff<pVVa-dA. 

We  may  arrive  at  some  interesting  theorems  by  assigning 
various  values  to  the  function  <p.     For  instance,  let 

<pdp  =  a  dp, 
then 

<p(VUvV)  =  <t'VUvVv'=-Ui>SV<t+V'S<t'Uv+SUpV(t, 

whence 

ffS^a-dv  =  ffV'Sa'dv+  fVadp. 
If 

<fdp  =  pSdpa, 


INTEGRALS  203 

then 

<pVUvV  =  pSUvVo-  -  VaUv, 
therefore 

ffV-adv  =  ffpSdvVa  -  fpSadp. 
If 

ipdp  =  pVdpa, 
tpVUvV  =  pV(VUvV)<r  -  SUva  +  aVv, 

therefore 

ffvdv  +  Sadv  =  -  ffpV(VUvV)<r  +  fpVdpa, 

hence 

2ffSadv  =  -  ffSp(VUvV)<r  +  fSpdpa. 

EXERCISES 

1.  Investigate  the  problems  of  article  19,  page  198,  as  to  the  applica- 
tion of  the  theorem. 

2.  Show  that  the  theorem  can  be  made  to  apply  to  a  line  which  is 
not  a  loop  by  joining  its  ends  to  the  origin,  and  after  applying  the 
theorem  to  the  loop,  subtracting  the  integrals  along  the  radii  from  0 
to  the  ends  of  the  line,  which  can  be  expressed  in  terms  of  dx,  along  a  line. 
Also  consider  cases  in  which  the  paths  follow  the  characteristic  lines  of 
Vadp  =  0. 

3.  The  theorem  may  be  stated  thus:  the  circulation  around  a  path 
is  the  total  normal  flux  of  the  curl  of  the  vector  function  a  through  the 
loop. 

4.  If  the  constant  current  la  amperes  flows  in  an  infinite  straight 
circuit  the  magnetic  force  H  at  the  point  p  (origin  on  the  axis)  is  for 

Tp<a        H  =  ^IVaP, 
and  for 

a<TP        H  =  0.2a?I/VaP, 

a  is  the  radius  of  the  wire.  Then  7vH  =  /(a/10)  inside  the  wire  and 
equals  zero  outside.  Integrate  H  around  various  paths  and  apply 
Stokes'  theorem.  In  this  case  the  current  is  a  vortex  field  of  intensity 
7ra27/10. 

5.  If  we  consider  a  series  of  loops  each  of  which  surrounds  a  given 
tube  of  vortex  lines,  it  is  clear  that  the  circulation  around  such  tube 
is  everywhere  the  same.  If  the  vector  <r  defines  a  velocity  field 
which  has  a  curl,  the  elementary  volumes  or  particles  are  rotating,  as 


204  VECTOR  CALCULUS 

we  have  seen  before,  the  instantaneous  axis  of  rotation  being  the  unit 
of  the  curl,  and  the  vector  lines  of  the  curl  may  be  compared  to  wires 
on  which  rotating  beads  are  strung.  It  is  known  that  in  a  perfect 
fluid  whose  density  is  either  constant  or  a  function  of  the  pressure  only, 
and  subject  to  forces  having  a  monodromic  potential,  the  circulation  in 
any  circuit  through  particles  moving  with  the  fluid  is  constant.  [Lamb, 
Hydrodynamics,  p.  194.]  Hence  the  vortex  tubes  moving  with  the  fluid 
(enclosing  in  a  given  section  the  same  particles),  however  small  in  cross- 
section,  give  the  same  integral  of  the  curl.  It  follows  by  passing  to  an 
elementary  tube  that  the  vortex  lines,  that  is,  the  lines  of  curl,  move 
with  the  fluid,  just  as  if  the  beads  above  were  to  remain  always  on  the 
same  wire,  however  turbulent  the  motion.  In  case  the  vortex  lines 
return  into  themselves  forming  a  vortex  ring,  this  leads  to  the  theorem 
in  hydrodynamics  that  a  vortex  ring  in  a  perfect  fluid  is  indestructible. 
It  is  proved,  too,  that  the  same  particles  always  stay  in  a  vortex  tube. 

6.  Show  that  for  a-  =  a(3S2otP  -  2SpP)  +  £(4#3/Sp  -  2SaP),  where 
Sa&  =  0,  the  integral  from  the  origin  to  2a  -J-  2/3  is  independent  of  the 
path.     Calculate  it  for  a  straight  line  and  for  a  parabola. 

7.  The  magnetic  intensity  H,  at  the  point  0,  from  which  the  vector 
p  is  drawn  to  a  filament  of  wire  carrying  an  infinite  straight  current  in 
the  direction  a,  of  intensity  I  amperes,  is  given  by 

H  =  0.27/Fap. 

Suppose  that  we  have  a  conductor  of  any  cross-section  considered  as 
made  up  of  filaments,  find  the  total  magnetic  force  at  0  due  to  all 
the  filaments.     Notice  that 

H  =  0.2/ Fa V  log  TVap, 

and  that  a  is  the  unit  normal  of  the  plane  cross-section  of  the  conductor. 
Hence 

ffHdA  =  ff0.2IVaV  log  TVapdA  =  f0.2I  log  TVapdp 

around  the  boundary  of  the  cross-section.  This  can  easily  be  reduced 
to  the  ordinary  form  0.21  j?  log  rdp.  This  expression  is  called  a  log- 
arithmic potential.  If  I  were  a  function  of  the  position  of  the  filament 
in  the  cross-section,  the  form  of  the  line-integral  would  change. 

For  a  circular  section  we  have  the  results  used  in  problem  4.  Con- 
sider also  a  rectangular  bar,  for  inside  points  and  also  for  outside  points. 

8.  If  or  and  r  are  two  vector  functions  of  p,  we  have  the  theorem 

SVUuVVo-t  =  St(VUvV)*  -  S<t(VUpV)t, 
whence 

ffSr(VUpV)o-  =  ffS<r{VUvV)r  +  fSdpar, 


INTEGRALS  205 

for  a  closed  circuit.  Show  applications  when  a  or  t  or  both  are  sole- 
noidal. 

9.  Show  that 

ffS-dvotS\7<x  =  fSdpaa  +  ffSdv(SaV)<r, 
ffS-Vuadv      =  JTuSadp  -  f  fuSV adp, 
ffS-X7uS7vdv=  fuSsjvdp  =  -  fvS\7udP, 
f1hiSVvdP=  [uv]ppl  -  f^vSS/udp. 

10.  Prove  Koenig's  theorems  and  generalize. 

(1)  Any  area  bounded  by  a  loop  generates  by  translation  a  volume 
=  —  Saw,  where  co  is  the  areal  axis; 

(2)  The  area  for  a  rotation  given  by  (a  +  Vap)at  is  —  J]  Saco  + 
ftScf  VpVpdp. 

21.  Green's  Theorem.  The  following  theorem  becomes 
fundamental  in  the  treatment  of  surface  integrals.  Refer- 
ring to  the  second  form  in  example  17,  page  189,  for  the 
expression  of  a  vector  in  the  vicinity  of  a  point,  which  is 

0"  =  FVSp[-  iV8pa0  +  iS8pVVdp<ro]  -  l&pSV<To 

we  see  that  if  we  multiply  by  a  vector  element  of  surface, 
Vdi8pd2dp,  and  take  the  scalar 

Scdrfpdidp  =  SUvVsp[]dA  -  iSV(r0Sdldpd2dp8p. 

If  now  we  integrate  over  any  closed  surface  the  first  term 
on  the  right  gives  zero,  since  the  bounding  curve  has  be- 
come a  mere  point,  and  thus,  indicating  integration  over 
a  closed  surface  by  two  J', 

j>  $&<jd\hpd<ihp  =  —  \S\7(TQjf  jfSdibpd'ibpbp. 

But  the  last  part  of  the  right  hand  member  is  the  volume 
of  an  elementary  triangular  pyramid  whose  base  is  given  by 
didpd28p.  Hence,  the  integral  is  the  elementary  volume  of 
the  closed  surface,  and  may  be  written  dv,  so  that  we  have 
for  an  elementary  closed  surface 

j> \fSad18pd2dp  =  SVvodv. 


206  VECTOR  CALCULUS 

If  now  we  can  dissect  any  volume  into  elements  in  which 
the  function  has  no  singularities  and  sum  the  entire  figure, 
then  pass  to  the  limit  as  usual,  we  have  the  important 
theorem 

ffS<rdlPd2p  =  fffSVv  dv. 

This  is  called  Greens  theorem,  or  sometimes  Green's  theorem 
in  the  first  form.  It  is  usually  called  Gauss'  theorem  by 
German  writers,  although  Gauss'  theorem  proper  was  only 
a  particular  case  and  Green's  publication  antedates  Gauss' 
by  several  years. 

The  theorem  may  be  stated  thus:  the  convergence  of  a 
vector  throughout  a  given  volume  is  the  flux  through  the 
bounding  surface. 

It  is  evident  that  we  can  generalize  this  theorem  as  we  did 
Stokes'  and  thus  arrive  at  the  generalized  Green's  theorem 
$ fQvdA  =  f  f  f$\/  dv.     v  is  the  outward  unit  normal. 

The  applications  are  so  numerous  and  so  important  that 
they  will  occupy  a  considerable  space. 

•  The  elementary  areas  and  volumes  used  in  proving  Stokes' 
and  Green's  theorems  are  often  used  as  integral  definitions 
of  convergence  or  its  negative,  the  divergence,  and  of  curl, 
rotation,  or  vortex.  For  such  methods  of  approach  see  Joly, 
Burali-Forti  and  Marcolongo,  and  various  German  texts. 

A  very  obvious  corollary  is  that  if  SVc  =  0  then 

$ \fSad1pd2p  =  0. 

It  follows  that  the  flux  of  any  curl  through  any  closed  sur- 
face is  zero.  Hence,  if  the  particles  of  a  vortex  enter  a 
closed  boundary,  they  must  leave  it.  Therefore,  vortex 
tubes  must  be  either  closed  or  terminate  on  the  boundary 
wall  of  the  medium  in  which  the  vortex  is,  or  else  wind 
about  infinitely.  We  may  also  state  that  if  SVa  =  0  the 
differential  expression  Sadipd2p  is  exact  in  the  sense  that 


INTEGRALS  207 

J%J%S(rdipd2p  is  invariant  for  different  diaphragms  bounded 
by  a  closed  curve,  noting  the  usual  restrictions  due  to 
singularities. 

We  proceed  to  develop  some  theorems  that  follow  from 
Green's  theorem.     Let  $Uv  be  —  pSUvcr,  then 

3>V  =  —  pSv<r  +  o- 
and  we  have 

fffadv  =  fffpSVvdv  -  ffpSUvadA. 

Let  $Uv  =  —  pVUva,  then  <i>V  =  —  pVVv  +  2a  and 
SSfvdv  =  ifffpVVvdv  -  \ffpVUvodA. 

Let  $Uv  =  pSpUva,  then  <J>V  =  pSpVv  +  Fpo-,  whence 
fffVpa  dv  =  -  fffp&Va  dv  +  ffpSpUvadA. 

Let  $17V  =  -  pVpVUixr, then $V  =  -  pFpFVo"  +  3PV, 

hence 

SSSVpadv  =  ifffpVpWadv-  \ffpVpVUvadA. 

Let  $E7V  =  SprUiHT,  then  3>V  =  SprV<r  +  Spa\/r  +  Sot, 

thence 

fffSar  dv  =  -  fffiSprV*  +  &rVr)<fo 

+  f  f&prTJva  dA. 

In  the  first  of  these  if  a-  has  no  convergence  we  have  the 
theorem  that  the  integral  of  cr,  a  solenoidal  vector,  through- 
out a  volume  is  equal  to  the  integral  over  the  surface  of  p 
multiplied  by  the  normal  component  of  a.  In  the  second 
we  have  the  theorem  that  if  the  curl  of  a  vanishes  through- 
out a  volume,  so  that  a-  is  lamellar  in  the  volume,  then  the 
integral  of  a  throughout  the  volume  is  half  the  integral 
over  the  surface  of  p  times  the  tangential  component  of  a 
taken  at  right  angles  to  a-.     In  the  third,  if  the  curl  of  cr 


208  VECTOR  CALCULUS 

vanishes  then  the  integral  of  the  moment  of  a  with  regard 
to  the  origin  is  the  integral  over  the  surface  of  Tp2  times  the 
component  along  p  of  the  negative  of  the  tangential  com- 
ponent of  a  taken  perpendicular  to  <r,  and  by  the  fourth 
this  also  equals  the  surface  integral  of  the  component 
perpendicular  to  p  of  the  negative  tangential  component  of 
<r  taken  perpendicular  to  a.  According  to  the  fifth  formula, 
if  a  solenoidal  vector  is  multiplied  by  another  and  the  scalar 
of  the  product  is  integrated  throughout  a  volume,  then  the 
integral  is  the  integral  of  —  SpaVr  throughout  the  volume 
-f-  the  integral  of  ScrprUv  over  the  surface. 

If  in  the  first,  second,  third,  and  fourth  we  set  c<t  for  a, 
and  in  the  fifth  ca  for  a  and  —  \<t  for  r,  we  have  from  the 
first  and  second  the  expression  for  X,  the  momentum  of  a 
moving  mass  of  continuous  medium,  of  density  c,  and  from 
the  third  and  fourth  the  moment  of  momentum,  /x,  and 
from  the  fifth  the  kinetic  energy.  If  the  medium  is  in- 
compressible, and  we  set  2k  =  V\/v,  since  SVca  =  0,  then 

X  =  fffcadv 
=  -  ffcpSUvadA  +  fffpSaVcdv 

+  SSfcpSV*  dv 

=  fffpcKdv+lfffpWcadv  -  \££cpVVvadA. 

ju  =  fffcVpadv 
=  ffcpSpUvadA  -  SSfcpSpVa  -  fffpSpVca 
=  UffcpVpK  +  \fffpVpWcadv 

-  \ffcpVpVVvadA. 
T  =  -  hSfSSa2cdv 

=  -  hffSpvUvac  dA  +  SffhcSpaVo-  dv 

+  hfffSpvVca  dv. 

In  case  c  is  uniform  these  become  still  simpler. 

If  we  set  a  =  S/u  and  r  =  \/w  in  the  above  formula  we 


INTEGRALS  209 

arrive  at  others  for  the  gradients  of  scalar  functions.  The 
curls  will  vanish.  If  further  we  suppose  that  u,  or  w,  or 
both,  are  harmonic  so  that  the  convergences  also  vanish 
we  have  a  number  of  useful  theorems. 

Othei  forms  of  Green's  theorem  are  found  by  the  follow- 
ing methods.     Set  $Uv  =  uS\7wUv,  then 

$V  =  u\/2w  +  SVuVw 

and  we  have  the  second  form  of  Green's  theorem  at  once 

SfS&VuVw  dv  =  ffuS\/wUv  dA  —  fffu\72wdv, 

and  from  symmetry 

yWSvWw  dv  =  ffwSVuUv  dA  —  fffw\/2u  dv. 

Subtracting  we  have 

J %J *J %(u\72w  —  w\72u)  dv 

=  ~  f£(,STJv[u\7w  -  wVu])dA. 

22.  Applications.  In  the  first  of  these  let  u  =  1,  then 
fffV2™  dv  —  —  J'.fSUj'VwdA.  If  then  w  is  a  har- 
monic function,  the  surface  integral  will  vanish,  and  if  V2w 
=  47Tju,  which  is  Poisson's  equation  for  potentials  of  forces 
varying  as  the  inverse  square  of  the  distance,  inside  the 
masses,  ju  being  the  density  of  the  distribution,  then 

ffSUvS7w  dA  =  ±ttM, 

where  M  is  the  total  mass  of  the  volume  distribution.  This 
is  Gauss'  theorem,  a  particular  case  of  Green's.  In  words, 
the  surface  integral  of  the  normal  component  of  the  force 
is  —  47r  times  the  enclosed  mass.  The  total  mass  is  l/4x 
times  the  volume  integral  of  the  concentration. 

In  the  first  formula  let  u  =  1/Tp  and  exclude  the  origin 


210  VECTOR  CALCULUS 

(a  point  of  discontinuity)  by  a  small  sphere,  then  we  have 

fffSV(l/Tp)Vwdv 

=  ffdA  SUrVw/TP  -  fffdv  V2w/Tp 

for  the  space  between  the  sphere  and  the  bounding  surface 
of  the  distribution  w,  and  over  the  two  surfaces,  the  normals 
pointing  out  of  the  enclosed  space.  But  for  a  sphere  we 
have  dA  =  Tp2  dw  where  co  is  the  solid  angle  at  the  center, 
and  dv  =  Tp2dwdTp.     Thus  we  have 

fffV2w/Tp  dv 

=  ffSdA  UuVw/Tp  -fffSv(l/TP)Vwdv 
=  ffSdA  UvVw/Tp  -fffSv(wV[l/TP])dv 

since  V2l/7p  =  0, 
=  ffSdA  UvVwjTp  -ffSdA  UvwV(l/Tp) 
=  ffSdA  UvVwjTp+ffSdA  VvVp\T2pw. 

Now  of  the  integrals  on  the  right  let  us  consider  first  the 
surface  of  the  sphere,  of  small  radius  Tp.  The  first  integral 
is  then  -  ffSUpX/wlTp-  T2pdco  =  -  ffSUpVw-  Tpda, 
and  if  we  suppose  that  the  normal  component  of  Vw,  that 
is,  the  component  of  Vw  along  p,  is  everywhere  finite,  then 
this  integral  will  vanish  with  Tp.  The  second  integral  for 
the  sphere  is  —  J?rf'SUpUpwT2pd(x)lT2p  =  —  tfj'wdu, 
and  the  value  of  w  at  the  origin  is  Wo,  then  this  integral  is 
47TWo  since  the  total  solid  angle  around  a  point  is  47r. 
Hence  we  have 

fffdv  V2w/Tp  =  ffSUv{\/wlTp  +  wUp/T2p)dA 

+  4twq 
and 

4x^o=  fffdvV2w/Tp 

-  ffSUp(Vw/Tp  -f  wUp/T2p)  dA, 


INTEGRALS  211 

where  the  volume  integral  is  over  all  the  space  at  which  w 
exists,  the  origin  excluded,  and  the  surface  integral  is  over 
the  bounding  surface  or  surfaces.  In  words,  if  we  know  the 
value  of  the  concentration  of  w  at  every  point  of  space, 
and  the  value  of  the  gradient  of  w  and  of  w  at  every  point 
of  the  bounding  surfaces  at  which  there  is  discontinuity, 
then  we  can  find  w  itself  at  every  point  of  space,  provided 
w  is  finite  with  its  gradient.  If  X72w  is  of  order  in  p  not 
lower  than  —  1  we  do  not  need  to  exclude  the  origin,  for 
the  integral  is  ///V2^  TpdcpdTp,  and  this  will  vanish 
with  Tp  when  V2w  is  not  lower  in  degree  than  —  1. 

EXERCISES 

1.  We  shall  examine  in  detail  the  problem  of  w  —  const,  over  a  given 
surface,  zero  over  the  infinite  sphere,  V2w  =  0  everywhere,  \/w  =  0 
on  the  inside  of  the  sphere,  but  not  zero  on  the  outside.  Then  for  the 
inside  of  the  sphere  the  equation  reduces  to 

4:irw0  =  -  £fwSUvUplT*pdA  =  4ttu;, 

hence  w  is  constant  throughout  the  sphere  and  equal  to  the  surface  value. 
On  the  outside  of  the  sphere,  we  have  to  consider  the  bounding  sur- 
faces to  be  the  sphere  and  the  sphere  of  infinite  radius,  so  that  we  have 

4^0  =  _  ffSdA  UvVw/Tp-  wffSdA  UuUpfTp2, 

where  the  first  integral  is  taken  over  both  surfaces  and  the  second 
integral  is  over  the  given  surface  only,  since  w  =  0  at  °° .  The  second 
integral  vanishes,  however,  since  it  is  equal  to  w  times  the  solid  angle 
of  the  closed  surface  at  a  point  exterior  to  it.  If  we  suppose  then  that 
\/w  is  0  at  «3  we  have  a  single  integral  to  evaluate 

4:irw0  =  —  j>  j> 'SdAU ri>\? 'w/T 'p  over  the  surface. 

A  simple  case  is 

—  SUv\/w  =  const.  =  C. 
Then 

4ttWo  =  CffdAITp. 

The  integration  of  this  and  of  the  forms  arising  from  a  different  assump- 
tion as  to  the  normal  component  of  V^  can  be  effected  by  the  use  of 
fundamental  functions  proper  to  the  problem  and  determined  by  the 
boundary  conditions,  such  as  Fourier's  series,  spherical  harmonics, 
and  the  like.     One  very  simple  case  is  that  of  the  sphere.     If  we  take 


212  VECTOR  CALCULUS 

the  origin  at  the  center  of  the  sphere  we  have  to  find  the  integral 

,f,fdA/T(P  -  Po) 

where  po  is  the  vector  to  the  point.  Now  the  solid  angle  subtended 
by  po  is  given  by  the  integral  —  r~lffdASpU{p  —  po)/T*(p  —  p0) 
and  equals  4t  or  0,  according  as  the  point  is  inside  or  outside  of  the 
sphere.  This  integral,  however,  breaks  up  easily  into  two  over  the 
surface,  the  integrands  being 

r-^T-Kp  -  po)  -  SpoU(P  -  P0)/T*(p  -  po), 

but  the  last  term  gives  0  or  —  47rr2/7Tp0,  as  the  point  is  inside  or  outside 
of  the  sphere.     Hence  the  other  term  gives 

ffdAlT{p  -  po)  -  47rr  or  4Trr2/Tp0 

as  the  point  is  inside  or  outside.     We  find  then  in  this  case  that 

w0  m  Cr2/Tpo. 

If  in  place  of  the  law  above  for  —  SUvS7w,  it  is  equal  to  C/T2(p  —  p0) 
we  find  that 

ffdAIT\P  -  po)  =  47rr/(r'  +  p02) 
or 

47^/(7^0  -  r^po). 
Inside 

_      r   r,A  S(p   ~   pp)(p   +  po) 

-  ffdA       TKp  -  po)        ' 

dA  =  27rr2  sin  Odd  =-  d[a2  +  r2  -  x2]  =  —xdx, 
a  a 

„po(p  —  po)  =  ax  cos  4/  _  a2  +  x2  —  r2 
T*(p  -  po)  "       x3  2x* 

ffdAS^f^=^fr+a'a+r(a^  +  l)dx  =  0 

T2(p  —  po)  aJr-a,a-r    \        X2  J 

or 

47TT2 

a 

The  differentiation  of  these  integrals  by  using  Vp0  as  operator  under 
the  sign  leads  to  some  vector  integrals  over  the  surface  of  the  sphere. 
2.  Show  that  we  have 

££UvdAIT(p  -  po)  =  |ttpo        or        |7rr3/^3Po-po 

for  inside  or  outside  points  of  a  sphere. 


INTEGRALS  213 

3.  Find  ffdAUu/T3(P  -  Po)  for  the  sphere. 

4.  Prove  f  fdAT^{p-fi)T-\p-oc)  =47rr/[(r2-a2)77(/S-«)]  or 

=  ^r2J[a(r2-a2)T(r2a-1  +0)]. 

5.  Consider  the  case  in  which  the  value  of  w  is  zero  on  a  surface 
not  at  infinity  but  surrounding  the  first  given  surface.  We  have  an 
example  in  two  concentric  spheres  which  form  a  condenser.  On  the 
inner  sphere  let  w  be  const.  =  Wi,  on  the  outer  let  w  =  0,  on  the  inner 
let  —  SUpVw  =  0,  inside,  =  Eh  outside,  on  the  outer  let  —  SUv\/w 
=  E2  on  the  inside,  =  Oon  the  outside. 

6.  If  w  is  considered  with  regard  to  one  of  its  level  surfaces,  it  is 
constant  on  the  surface,  and  the  integral  —  £  f  SdAU  vU  p\T2  pio 
becomes  for  any  inside  point  4:irw,  hence  we  have 

4irw0  -  A.™  =  fffdv\72wlTP  -  £ £SdAUuVw/TP. 

If  then  w  is  harmonic  inside  the  level  surface,  it  is  constant  at  all  points 
and 

47r(w0  -  to)  m  -  £fSdAUv\7wlTp. 

But  since  w0  is  constant  as  we  approach  the  surface,  V^o  =0,  and 
V(w  —  Wo)  =  0,  so  that  X7w  =  0.  Hence  w0  =  w.  If  w  vanishes  at 
oo  and  is  everywhere  harmonic  it  equals  zero. 

7.  If  two  functions  Wi,  w2  are  harmonic  without  a  given  surface, 
vanish  at  » ,  and  have  on  the  surface  values  which  are  constantly  in  the 
ratio  X  :  1,  X  a  constant,  then  W\  =  \W2. 

8.  If  the  surface  Si  is  a  level  for  both  the  functions  u  and  w,  as  also 
the  surface  S2  inside  Si,  and  if  between  Si  and  $2,  u  and  w  are  harmonic, 
then 

(U  —  Ui)(w2   —  Wi)    =   (W  —  Wi)(ll2  —  Ui). 

For  if  w  =  <p(u),  then  V2w  =  0  =  <p"(u)T2\7u,  hence  <p(u)  —  au  +  b, 
etc. 

[A  scalar  point  function  w  is  expressible  as  a  function  of  another 
scalar  function  u  if  and  only  if  V\/w\7u  =  0.] 

9.  Outside  a  closed  surface  S,  Wi  and  w2  are  harmonic  and  have  the 
same  levels.  Si  vanishes  at  •  while  w2  has  at  00  everywhere  the  con- 
stant value  C.     Then  w2  =  Bwi  +  C. 

For  Vw2  =  tVwh  V2w2  =  V^V^i  =  0,  thus  V*  =  0,  or  V^i  =  0, 
and  t  =  B  or  wi  =  const. 

10.  There  cannot  be  two  different  functions  W\,  w2  both  of  which 
within  a  given  closed  surface  are  harmonic,  are  continuous  with  their 
gradients,  are  either  equal  at  every  point  of  S  or  else  SUvX/Wi  =SUv\/w2 
at  every  point  of  S  while  at  one  point  they  are  equal. 

Let  u  =  Wi  —  w2,  then  V2w  =  0,  SJu  =  0  on  S  or  else  SUv\/u  =  0, 
and  at  one  point  Vw  =  0. 


214  VECTOR  CALCULUS 

11.  Given  a  set  of  mutually  exclusive  surfaces,  then  there  cannot  be 
two  unequal  functions  w\,  Wi,  which  outside  all  these  surfaces  are 
harmonic,  continuous  with  their  gradients,  vanish  at  <»  as  Tp~l,  their 
gradients  vanishing  as  Tp-2,  and  at  every  point  of  the  surfaces  either 
equal  or  SUvVwi  =  SUvVwt- 

23.  Solution  of  Laplace's  Equation.  The  last  problems 
in  the  preceding  application  show  that  if  we  wish  to  invert 
V2w  =  0,  all  we  need  are  the  boundary  conditions,  in  order 
to  have  a  unique  solution.  In  case  V2u  is  a  function  of 
P>f(p)>  we  can  proceed  by  the  method  of  integral  equations 
to  arrive  at  the  integral.  However  the  integral  is  express- 
ible in  the  form  of  a  definite  integral,  as  well  as  a  series, 

w0  =  l/4:w[fSSdvV2w/Tp 

-  ffSUviVw/Tp  +  wUp/T2p)dAl 

The  first  of  these  integrals  is  called  the  potential  and  written 
Pot.     Thus  for  any  function  of  p  whatever  we  have 

Vot  q,  =  fffqdvlT(p-  pQ) 

where  p  describes  the  volume  and  p0  is  the  point  for  which 
Pot  qo  is  desired.  Let  Vo  be  used  to  indicate  operation  as 
to  po,  then  we  have 

Vo  Pot  g0  =  VoffSqdv/T(p  -  p0) 

=  fff[dvU(p  -  p0)/r2(p  -  Po)]q 
-  -SSfV[qlT(p-  p0)]dv 

+  SffdWq/T(p-  po) 
=  Pot  Vg  -  ffdAUvqlT(p  -  Po). 

If  we  operate  by  Vo  again,  we  have 

Vo2  Pot  q0  =  Pot  V2?  -  ffdA[Uv\7qlT(p  -  po) 

+  V'Uvq/T'(p  -  po)]. 

But  the  expression  on  the  right  is  4x^0,  hence  we  have  the 


INTEGRALS  215 

important  theorem 

Vo2  Pot  q0  =  4:irq0. 

That  is,  the  concentration  of  a  potential  is  4x  times  the 
function  of  which  we  have  the  potential.  In  the  case  of  a 
material  distribution  of  attracting  matter,  this  is  Poisson's 
equation,  stating  that  the  concentration  of  the  potential 
of  the  density  is  4r  times  the  density;  that  is,  given  a 
distribution  of  attracting  masses,  they  have  a  potential  at 
any  given  point,  and  the  concentration  of  this  potential  at 
that  point  is  the  density  at  the  point  -5-  4-7T. 

The  gradient  of  Pot  q0  was  called  by  Gibbs  the  Newtonian 
of  g0,  when  the  function  q0  is  a  scalar,  and  if  q0  is  a  vector, 
then  the  curl  of  its  potential  is  called  the  Laplacian,  and  the 
convergence  of  its  potential  is  called  the  Maxwellian  of  q0. 
Thus 

New  q0  =  Vo  Pot  P,        Lap  <r0  =  V\/o  Pot  <r0, 
Max  (To  =  £  Vo  Pot  co. 

We  have  the  general  inversion  formula 

47rVo~2Vo2?  =  47rgo 

-  SSfV2q/T(p  -  Po)dv 

-  ffdA[UvTqlT{p  -  p0) 

+  U(p  -  p0)qUr/T*(p  -  p0)J. 

This  gives  us  the  inverse  of  the  concentration  as  a  potential, 
plus  certain  functions  arising  from  the  boundary  conditions. 
We  may  also  define  an  integral,  sometimes  useful,  called 
the  Helmholtzian, 

Him.  Q  m  fffQT{p  -  Po)dv. 

Certain  double  triple  integrals  have  been  defined: 

Pot  0,  v)  =  ffffffu(p1)v(p2)dv1dv2/T(p1  -  p2), 


216  VECTOR  CALCULUS 

Pot  (to)  =  fffffS  -  Sh  dvidvJTfa  -  p2), 
Lap  (to)  =  ffffff  +  5to(Pi  -  p2)^i^2/P(Pl-p2), 
New  («,  f)  =  SSSSSS-S{i(pi-p2)vldvidv2IT'(pi-pt), 
Max(£,*)  =  -  ffffffvlSUpi-P2)dvldv2ir(Pl-P2). 

EXERCISES 

1.  Iff  =  —  VP  is  a  field  of  force  or  velocity  or  other  vector  arising 
from  a  scalar  function  P  as  its  gradient,  then 

Po  =  -  SSSSV£dv/[4irT(P-  po)]  +  ffdA[SUvll&*T{p  -  po)) 

+  PC/yV^(p-po)/47r]. 

If  P  is  harmonic  the  first  term  vanishes,  if  £  =  0  the  first  two  vanish. 

2.  If  £  =  V<r,  that  is,  it  is  a  curl  of  a  solenoidal  vector, 

°o  =  fffVV*  dv/[4irT(p  -  po)]  -  f<fdA[VUv<rl[±TcT{p  -  po)] 

+  U(p  -  p9)<rlU,[4*T*(j>  -  po)]. 

3.  We  may,  therefore,  break  up  (in  an  infinity  of  ways)  any  vector 
into  two  parts,  one  solenoidal  and  the  other  lamellar. 

Thus,  let  a  =  7T  +  t  where  £v  r  ■  0,  and  Wir  =  0,  then  Sv  <r  =  SVx 
and  since  VVt  =  0,  this  may  be  written  Vt  =  &Vo"  whence  x. 

VVc  =  FVt  =  Vr  whence  t.  We  have,  therefore,  from  these  two 
47T<r0  =  VfffSdvV<r/T(p  -  Po)  -  V £  £SdAUvalT{p  -  Po) 

+ V  jffPSdA  UvV  (UT(p  -  po)  +  V  V  SSfVV*dv/T(p  -  Po) 
-WffVU*adA/T(p-po)  +  VVffDSU*S7(p+(P-po)dAt 

where  P  is  such  that  V2P  =  *S'V<r  and  D  such  that  \72D  =  Vs/a. 

3.  Another  application  is  found  in  the  second  form  of  Green's 
theorem.     According  to  the  formula 

SffiyW1™  -  wV2u)  dv  =  -  tf£(SUv[u\7w  -  w\/u])dA 

it  is  evident  that  if  G  is  a  function  such  that  V2G  =  0,  and  if,  further, 
G  has  been  chosen  so  as  to  satisfy  the  boundary  condition  SUvS7G  =  0, 
then  the  formula  becomes 

SffGs^wdv  =  -  ££SUv\7wGdA. 

If  then  V2w  is  a  given  function  we  have  the  integral  equation 

JfGSUvVwdA  -  -  fffGj{P)dv. 

Similar  considerations  enable  us  to  handle  other  problems. 

4.  If  u  and  w  both  satisfy  V2/  =  0,  then  we  have  Green's  Reciprocal 
Theorem: 

ffuSUvSfw  dA  =  ffwSUvVudA, 


Thus  let 
therefore 


INTEGRALS  217 


ff  ^p  dA  =  ffuSUvV  -L  dA. 


5.  Let  A  relate  to  a  as  V  to  p;  then 

A  Pot  Q  =  ff/QdvU(p  -  a)/T*(P  -  a) 

=  fffV(Q/T(p  -  cc))dv  +  fffdWQ/T(p  -  a) 

=  Pot  VQ  -  ffdAUuQ/T(P  -  a). 

If  Q  —  0  on  the  surface,  the  surface  integral  =  0. 

New  P  =  Pot  V  -  ffVvP  dAjT{p  -  a)  =  A  Pot  P  when  Pot  exists. 

Lap  a  =  V  Pot  Vo-  -  ££VTJvadA\T{p  -  a)  =  VA  Pot  a  when  Pot 
exists. 

Max  o-  =  S  Pot  V<r  -  £  fSUvadAITip  -a)  =  SA  Pot  o-  when  Pot 
exists. 

A2  PotQ  =  Pot  V2Q  -  ££Uv\7QdAIT(p  -  a) 

+  //diVi[^/77i(P  -«)]. 
If  Q  =  0  on  the  surface,  that  is,  if  Q  has  no  surface  of  discontinuity, 

A2  Pot  Q  -  Pot  V2Q, 

A  New  P  =  A2  Pot  P, 

A  Lap   o-  =  A7A  Pot  a, 

A  Max  a   =  A/SA  Pot  <r. 

6.  If  j8  is  a  function  of  the  time  t,  then 


d--^[yy/i(rvFv^+M^J 


t+br 


r  r 

+  VV  £<fj  Vdu  pt+br  -  ff  Vdv  WPt+br 

where  the  subscript  means  t  +  br  is  put  for  t  after  the  operations  on  0 
have  occurred. 


15 


CHAPTER  IX 

THE  LINEAR  VECTOR  FUNCTION 

1.  Definition.  If  there  is  a  vector  a  which  is  an  integral 
rational  function  <p  of  the  vector  p, 

a  =  <p'P, 

and  if  in  this  function  we  substitute  for  p  a  scalar  multiple 
tp  of  p,  then  we  call  the  vector  function  a  linear  vector  func- 
tion if  a  becomes  ta  under  this  substitution.  It  is  also  called 
a  dyadic. 

The  function  <p  transforms  the  vector  p,  which  may  be 
in  any  direction,  into  the  vector  <r,  which  may  not  in  every 
case  be  able  to  take  all  directions.  If  p  =  a,  then  we  have 
(pp  =  <pa,  and  <p  as  an  operator  has  a  value  at  every  point 
in  space.  We  may,  in  fact,  look  upon  <p  as  a  space  trans- 
formation that  deforms  space  by  a  shift  in  its  points  leaving 
invariant  the  origin  and  the  surface  at  infinity.  In  the 
case  of  a  straight  line 

Vap  =  /?,         or         p  =  xa  +  cTl(5, 

we  see  that  the  operation  of  <p  on  all  its  vectors  gives 

a  =  x<pa  +  (pVa~1^f 

and  this  is  a  straight  line  whose  equation  is 

Vipaa  =  V<pa(pVa~1^, 

which  will  later  be  shown  to  reduce  to  a  function  of  (3 
only,  <p(3.     Hence  <p  converts  straight  lines  into  straight 
lines.     The  lines  a  for  which  Vacpa  =  0,  remain  parallel 
218 


THE   LINEAR  VECTOR   FUNCTION  219 

to  their  original  direction,  others  change  direction.  Again 
if  we  consider  the  plane  S-afip  =  0  or 

p  =  xa  +  y(3,     ,    a  =  x<pa  +  y<pf}, 

so  that 

S(r<pct<pp  =  0. 

Hence  planes  through  the  origin,  and  likewise  all  planes, 
are  converted  into  planes.  These  will  be  parallel  to  their 
original  direction  if  Va(3  =  uV<pa<p(3,  or 

VVa$V(pa<p&  =  0  =  Scx<pa<p(3=  S(3<pa<p(3=  Sa(3<pa  =  So@<pP. 

Now  Va(3  is  normal  to  the  plane,  and  /3  is  any  vector  in  the 
plane,  and  <p(3  by  the  equation  is  normal  to  Vafi,  hence 
<p(3  =  va  +  w(3  for  all  vectors  0  in  the  plane. 

Since  <p0  =  0,  the  function  leaves  the  origin  invariant. 
Consequently  the  lines  and  planes  through  the  origin  that 
remain  parallel  to  themselves  are  invariant  as  lines  and 
planes.  These  lines  we  will  call  the  invariant  lines  of  <p, 
and  the  planes  the  invariant  planes  of  tp. 

2.  Invariant  Lines.  In  order  to  ascertain  what  lines  are 
invariant  we  solve  the  equation 

Va<pa  =  0,         or         (pa  =  ga, 

that  is 

(tp  -  g)a  =  0. 

First  we  write  a  in  the  form 

aS\fiv  =  \SfJiva  +  ixSvka  +  vSXfxa, 

where  X,  ju,  v  are  any  three  noncoplanar  vectors.  Then  we 
have  at  once 

(<p  —  g)\Sixvicx  +  (<p  —  g)nSv\a  +  (<p  —  g)pS\fxa  =  0. 


220  VECTOR  CALCULUS 

But  this  means  that  we  must  have  for  any  three  non- 
coplanar  vectors  X,  /i,  v 

S(<p  -  g)\(<p  -  g)fi(<p  -  g)v  =  0 

=  tfSXiiv  —  g2(S\ii<pv  +  S\<ptxi>  +  S<p\nv) 

+  g(S\(pfJL(pV  +  S\jJl<pV  +  S(f\(pfJLP)  —  S<p\<piJ.<pi>, 

an  equation  to  determine  g,  which  we  shall  write 

gz  -  mig2  +  m2g  -  m3  =  0, 
called  the  /a<6n<  equation  of  #>,  where  we  have  set 

Wl  =    (S\jA<pV  +  S\<pflP  +  S<p\fAl>)/S\fJLl>, 

rri2  =  (S\(pii(pp  -+-  S<pkyupv  +  S(p\<piJLp)lS\fjLv, 

These  expressions  are  called  the  nonrotational  scalar  in- 
variants of  <p.  That  they  are  invariant  is  easily  seen  by 
substituting  X'  +  v/jl  for  X.  The  resulting  form  is  precisely 
the  same  for  Xr,  ju,  p,  and  from  the  symmetry  involved  this 
means  that  for  X,  /x,  v  we  can  substitute  any  other  three 
noncoplanar  vectors,  and  arrive  at  the  same  values  for 
mi,  m2,  m3.  It  is  obvious  that  m3  is  the  ratio  in  which  the 
volume  of  the  parallelepiped  X,  jjl,  v  is  altered.  If  m3  =  0 
one  or  more  of  the  roots  of  the  cubic  are  zero.  The  number 
of  zero  roots  is  called  the  vacuity  of  (p.  If  is  obvious  that 
the  latent  cubic  has  either  one  or  three  real  roots. 

3.  General  Equation.  We  prove  now  a  fundamental 
equation  due  to  Hamilton.  Starting  with  <p  we  iterate  the 
function  on  any  vector,  as  p,  writing  the  successive  results 
thus 

p,  <pp,  <p<pp  =  <p2p,         <p<p<pp  =  <p<p2p  =  <p3p,  "•. 

We  have  then  for  any  three  vectors  X,  ,u,  v  that  are  not 
coplanar 


THE   LINEAR   VECTOR   FUNCTION  221 

S\pi>(<p3p  —  mi<p2p)  =  (p2(<pp  —  m\p)S\pv 

=  <p2[<p\Spvp  -\-  •  -  -  —  pSpv(p\  —  •  •  •] 

=  -  <p2[V-VppV<p\p+  •••] 

=  <p\V'V<p\pVp.v+  •••] 

=  <p[<p\Sv<ppp  +  <pp<S\<pvp  +  <pvSp<p\p 

—    <p\SpL(pvp  —    (fpSvcpXp 

—  <pi>S\(pfxp]. 
Adding  to  this  result  S\pu> -m%ipp,  we  have 

S\pv((p3p  —  mnp2p  +  m<npp) 

=  <p[\S<pfi<pvp  +  pSipVipkp  +  vS<p\(pp,p]  =  pS(f\cppapv. 

Subtracting  SXfMV-rritp  from  both  sides  and  dropping  the 
nonvanishing  factor  S\p,i>,  we  have  the  Hamilton  cubic  for  <p 

<psp  —  mi<p2p  +  m*<pp  —  mzp  =  0. 

This  cubic  holds  for  all  vectors  p,  and  hence,  may  be  written 
symbolically 

<p3  —  mnp2  +  m2(p  —  ra3  =  0 

identically.  This  is  also  called  the  general  equation  for  <p. 
It  is  the  same  equation  so  far  as  form  goes  as  the  latent 
equation.     Hence  we  may  write  it  in  the  form 

(<p  -  gi)(<p  —  g*)(<p  —  gz)  =  0. 

In  other  words,  the  successive  application  of  these  three 
operators  to  any  vector  will  identically  annul  it. 

We  scarcely  need  to  mention  that  the  three  operators 
written  here  are  commutative  and  associative,  since  this 
follows  at  once  from  the  definition  of  linear  vector  operator, 
and  of  its  powers. 

It  is  to  be  noted,  too,  that  <p  may  satisfy  an  equation  of 
lower  degree.  This,  in  case  there  is  one,  will  be  called  the 
characteristic  equation  of  <p.     Since  <p  must  satisfy  its  general 


222  VECTOR   CALCULUS 

equation,  the  process  of  highest  common  divisor  applied 
to  the  two  will  give  us  an  equation  which  <p  satisfies  also, 
and  as  this  cannot  by  hypothesis  be  lower  than  the  char- 
acteristic equation  in  degree  and  must  divide  it,  it  is  the 
characteristic  equation.  Hence  the  factors  of  the  char- 
acteristic equation  are  included  among  those  of  the  general 
equation.  We  proceed  now  to  prove  that  the  general 
equation  can  have  no  factors  different  from  the  factors 
of  the  characteristic  equation. 

(1)  Let  the  characteristic  equation  be 

(<p  -  g)p  =  0 

for  every  vector;  then  assuming  any  X,  /x,  v,  we  find  easily 
for  the  latent  equation 

x*-Sgx2+3g2x-g*=  0, 

so  that  the  general  equation  is 

(cp  -  gf  =  0. 
In  this  case 

if  =  [g\SM)  +  gpSrkQ  +  gpSlnOV&V, 

where  X,  /z,  v  are  given  for  a  given  <p. 

(2)  Let  the  characteristic  equation  be 

(<P  -  9i)(<P  -  92) P  =  0, 

then  by  hypothesis,  there  is  at  least  one  vector  a  for  which 
we  have 

(<p  -  gi)a  +  0, 

and  at  least  one  fi  for  which 

(<p  -  gt)0  4=  0. 
Let  us  take  then 

O  -  gi)a  =  X,         (<p—  g2)(S  =  M- 


THE   LINEAR  VECTOR  FUNCTION  223 

Then 

(<p  -  g2)\  =  0,         (<p-  gi)fx  =  0. 

Hence,  we  cannot  have  X  and  ju  parallel,  else  gi  =  g2,  which 
we  assume  is  not  the  case,  since  from 

(<p-  g2)U\  =  0,        (<p-  g1)Ufx=  0, 
we  have 

g2U\  =  giUn,        and        g2  =  gu 

if  X  is  parallel  to  /z,  that  is  if  U\  would  =  Up, 

There  is  still  a  third  direction  independent  of  X  and  /z, 
say  v.     Let 

cpv  =  av  +  bjjL  +  cX. 
Then  we  have 

(<p  -  ft)*  =  (a  -  gi)j>+  bfx  +  cX. 
Since 

(<p-  fc)(*  -  9i)v  =  0, 
(a  -  gx)(<p  -  g2)v  —  b(g2  -  fi)p  =  0 

=  (a—  gY){a  —  g2)v  +  6 (a  -  g2)fx  +  c(a  —  g{)\. 

We  must  have,  therefore,  either 

a  =  gi        and        6=0, 
or 

a  =  g2        and        c  =  0. 

As  the  numbering  of  the  roots  is  immaterial,  let  us  take 
a  =  git  b  =  0,  then 

<pv  =  giv  +  cX,         <pX  =  #2X,         ^>m  =  9iV> 

We  notice  that  if  c  #  0,  we  can  choose  v'  =  v  —  (cjg2)\, 
whence  ipv'  =  giv'  and  we  could  therefore  take  c  =  0. 
Hence 

g3  -  g\2gi  +  g2)  +  ^(2fir^2  +  gY2)  -  g?g2  =  0, 
<p  =  [guiS\vQ  +  givSXpQ  +  #2X£mK)]ASX/xj>, 


221  VECTOR  CALCULUS 

and  the  general  equation  is 

(<P  -  9i)2(<P  ~  92)  =  0. 
(3)  Let  the  characteristic  equation  be 
(<p  -  g)*p  =  0. 
Then  there  is  one  direction  X  for  which  we  have 

<p\  =  g\, 

and  there  may  be  other  directions  for  which  the  same  is 
true.     There  is  at  least  one  direction  \i  such  that 

(cp  -  g)fi  =  X. 
We  have,  therefore, 

<PV  =  g»  +  X        <?X  =  g\. 
Let  now  v  be  a  third  independent  direction,  then  we  have 

(pv  =  av  +  bjj.  +  cK, 
(<p  -  g)v  =  (a  —  g) v  +  6/x  +  c\, 
(<p  -  gfv  =  0  =  (a  -  gfv  +  b(a  -  g)p  +  [b  +  c(a  -  g)]K. 

Therefore,    we    have    a  =  g,    6=0,    <pp  =  gv  +  cX   and 
<£>(*>  —  c/x)  =  g{y  —  c/jl)  =  gv' ,  and  the  general  equation 

i*  -  g)3  =  0, 

<p  =  g  +  XSj/XO/SX/x*'. 

We  are  now  in  a  position  to  say  that  the  general  equation 
has  exactly  the  same  factors  as  the  characteristic  equation. 
Further  we  can  state  as  a  theorem  the  following: 

(a)  //  the  characteristic  equation  is  of  first  degree, 

O  -  gY)p  =  0, 

then  every  vector  is  converted  into  g\  times  that  vector,  by  the 
operation  of  (p. 


THE   LINEAR   VECTOR  FUNCTION  225 

(6)  //  the  characteristic  equation  is  of  the  form 
O  -  9i)(<P  ~  92)  =  0, 

then  there  is  one  direction  X  such  that  <pk  =  92K,  while  for 
every  vector  in  a  given  plane  of  the  form  x\i.-\-  yv  we  have 

(<p-  #i)Om  +  yv)  =  0. 

Hence  <p  multiplies  by  gi  every  vector  in  the  plane  of  /a,  v, 
and  by  g2  all  vectors  in  the  direction  X. 
(c)  If  the  characteristic  equation  is 

W  -  g,f  =  0, 

there  is  a  direction  such  that 

<p\  =  gi\ 

and  a  given  plane  such  that  for  every  vector  in  it  x\x-\-  yv 
we  have 

(<P  —  9i)(w  +  yv)  =  ^X. 

If  (<P  —  gi)v  =  v\  (<P  —  gi)v  =  w\  we  may  set 

• 
w 

giving  (p/i  =  gip.     Therefore  <p  extends  all  vectors  in  the  ratio 
gi,  and  shears  all  components  parallel  to  v  in  the  direction  X. 
4.  Nondegenerate  Equations.     We  have  left  to  consider 
the  three  cases 

(<p  —  9i)(<p  —  92) (<p  -  gz)  =  0, 

O  -  gi)2(<P  -  02)  =  0, 

(v  -  g,f  =  0. 

In  the  last  case  we  see  easily  that  there  is  a  set  of  unit 
vectors  X,  ju,  v  such  that 


226  VECTOR  CALCULUS 

<p\  —  g{K  +  mo, 

<PH  =  giii  +  vb, 

<pv  =  giv. 
Hence  we  see  that 

<p(x\  +  yi*  +  zv)  =  gi{x\  +  y\x  +  zv)  +  a*M  +  6y*> 
=  gi(x\  +  2/M  +  •*)  +  a(a*M  +  0*0 

+  (6  -  a)yy, 
<p(x»  +  yv)  =  gi(xn  +  ?/*>)  +  fo^, 

<p  =  gi  +  [apSuvQ  +  bvSv\Q]/S\nv. 

Therefore  <p  extends  all  vectors  in  the  ratio  g\,  shears  all 
vectors  X  in  the  direction  of  m>  and  all  vectors  /x  in  the 
direction  v. 

In  the  first  case  we  see  that  there  is  at  least  one  vector 
p  such  that 

{<P  -  9\){<P  -  9s) P  =  A, 
where 

<p\  =  g{K. 

Likewise  there  are  vectors  that  lead  to  /x  and  v  where 
<PH  =  g2n,  <pp  =  gzv.  These  are  independent,  and  there- 
fore if  we  consider  any  vector 

p  =  x\  +  yn  +  zp, 
we  have 

<pp  =  xg{K  +  2/02M  +  zg3v, 
<p  =  [g^SfivQ  +  fwJSrhO  +  gsvS\nO]lS\fiP. 

Evidently  we  can  find  X,  /x,  v  by  operating  on  all  vectors 
necessary  in  order  to  arrive  at  nonvanishing  results  by 

(<P  —  9z)(<P  —  9s),      (<P  —  9i)(<P  —  9*)>      (<P  —  9\)(<P  —  9t) 

respectively. 

In  the  second  case,  we  see  in  a  similar  manner  that  there 


THE   LINEAR  VECTOR  FUNCTION  227 

are  three  vectors  such  that 

£>X  =  g{K  +  \x,         <pfi  =  giii,         <pv  =  g2p, 
<P  =  IgiQiSpvQ  +  nSvkQ  +  fiwStoQ  +  jtSMW&Vv. 

5.  Summary.  We  may  now  summarize  these  results  in 
the  following  theorem,  which  is  of  highest  importance. 

Every  linear  vector  function  satisfies  a  general  cubic,  and 
may  also  satisfy  an  equation  of  lower  degree  called  the  char- 
acteristic equation.  If  the  equation  of  lowest  degree  is  the 
cubic,  then  it  may  have  three  distinct  latent  roots,  in  which 
case  there  corresponds  to  each  root  a  distinct  invariant  line 
through  the  origin,  any  vector  in  each  of  the  three  directions 
being  extended  in  a  given  ratio  equal  to  the  corresponding  root; 
or  it  may  have  two  equal  roots,  in  which  case  there  corresponds 
to  the  unequal  root  an  invariant  line,  and  to  the  multiple  root 
an  invariant  plane  containing  an  invariant  line,  every  vector 
in  the  plane  being  multiplied  by  the  root  and  then  affected  by 
a  shear  of  its  points  parallel  to  the  invariant  line  in  the  plane; 
or  there  may  be  three  equal  roots,  in  which  case  there  is  an 
invariant  line,  a  plane  through  this  line,  every  line  of  the 
plane  through  the  origin  being  multiplied  by  the  root  and  its 
points  sheared  parallel  to  the  invariant  line,  and  finally  every 
line  in  space  not  in  this  plane  is  multiplied  by  the  root  and 
its  points  sheared  parallel  to  the  invariant  plane.  In  case 
the  function  satisfies  a  reduced  equation  which  is  a  quadratic, 
this  quadratic  may  have  unequal  roots,  in  which  case  there 
is  an  invariant  line  corresponding  to  one  root  and  an  invariant 
plane  corresponding  to  the  other,  any  line  in  the  plane  through 
the  origin  being  multiplied  by  the  corresponding  root;  or  there 
may  be  two  equal  roots,  in  which  case  there  is  an  invariant 
plane  such  that  every  line  in  the  plane  is  multiplied  by  the 
root  and  every  vector  not  in  the  plane  is  multiplied  by  the  root 
and  its  points  displaced  parallel  to  an  invariant  line.     In  case 


22$  VECTOR  CALCULUS 

the  reduced  equation  is  of  the  first  degree,  every  line  is  an 
invariant  line,  all  vectors  being  extended  in  a  fixed  ratio. 
Where  there  are  displacements,  they  are  proportional  to  the 
distance  from  the  origin,  and  the  region  displaced  is  called  a 
shear  region. 

Hence  <p  takes  the  following  forms  in  which  gif  g2,  gz  may- 
be equal,  or  any  two  may  be  equal: 

I.     [g&SpyQ  +  g2pSya()  +  g^ySapOVSapy;     reduced 

equations  for  gx  =  g2  or  gx  =  g2  =  #3; 
II.     [9laSPyQ  +  giPSyaQ  +  mScfiO  +  a0Sj8y()]/So0y; 

reduced  equation  for  gi  =  g2,  or  if  a  —  0; 
III.     g  +  [(a/3  +  cy)80yQ  +  bySya ()]/SaPy,  reduced  if 
a  =  0  =  c,  or  a  =  0  =  b  =  c. 

EXAMPLES 
(1).  Let    <pp=V-app,    where    SaP  +  0.     Take    X  =  a, 
u  =  P,  v  =  Vafi,  then  we  find  with  little  trouble 

mi  =  -  Sap,        m2  =  -  a2/?2,         ra3  =  a2p2SaP, 

and  the  characteristic  equation  of  <p, 

(tp  +  Sap)(<p  -  Tap)(<p  +  Tap)  =  0. 

Hence  there  are  three  invariant  lines  in  general,  and  oper- 
ating on  p  by  (<p  +  Safi)(<p  —  TaP),  we  find  the  invariant 
line  corresponding, 

(<p  +  SaP)p  =  aSpp  +  pSap, 
(<p-  TaP)(<p  +  Sap)p 

=  a2pSPp  +  P2aSap  -  aTaPSPp  -  pTaPSap 
=  -  (TaSpp+  TpSap)(Ua+  UP)Tap. 

Hence  the  invariant  line  corresponding  to  the  root  TaP  is 
Ua  +  Up.     The  other  two  are 

Ua  -  Up        and         UVap. 


THE   LINEAR   VECTOR   FUNCTION  229 

(2).  Let  <pp  =  Vafip. 

(3).  Let  <pp  =  g2aS(3yp  +  frtfSyap  +  ySo&p)  +  hfiSaPp. 

(4).  Let  <pp=  gp+  (fifi  +  ly)Sfap  +  r(3Syap. 

(5).  Let  <pp  =  Vep. 

6.  Solution  of  cpp  =  a.     It  is  obvious  that  when  <p  satis- 
fies the  general  equation 

<p3  —  mi<p2  +  m2(p  —  ra3  =  0,  ra3  4=  0, 

then  the  vector 

m%<p~lp  =  (w2  —  miv?  +  <^2)p. 

For  if  we  take  the  <p  function  of  this  vector,  we  have  an 
identity  for  all  values  of  p.  Also  this  vector  is  unique,  for 
if  a  vector  a  had  to  be  added  to  the  left  side,  or  could  be 
added  to  the  left  side,  then  it  would  have  to  satisfy  the 
equation  <pa  =  0.  But  if  ra3  4=  0,  there  is  no  vector  satis- 
fying this  equation,  for  this  equation  would  lead  to  a 
zero  root  for  <p.  Hence,  if  cpp  =  X,  ra3p  =  m2X  —  mnp\  +  <p2X, 
which  solves  the  equation. 

If  <p  satisfies  the  general  equation 

(pi  —  mnp2  +  m<2<p  =  0,  m%  #=  0, 

then  we  have  one  and  only  one  zero  root  of  the  latent  equa- 
tion, and  corresponding  to  it  a  unique  vector  for  which 
<pa  =  0,  and  if  (pp  =  X, 

m2p  =  xa  +  [m\(p  —  (p2)p  =  xa  +  w&iX  —  <pX. 

If  (p  satisfies  the  cubic 

(pz  —  rriiv2  =0,  mi  4  0, 

the  vacuity  is  two,  and  we  have  two  cases  according  as 
there  is  not  a  reduced  equation,  or  a  reduced  equation  exists 


230  VECTOR  CALCULUS 

of  the  form  <p2  —  m\<p  =  0.  In  either  case  the  other  root 
is  mi.  There  is  a  corresponding  invariant  line  X,  and  if  the 
vector  a  is  such  that  <pa  =  0,  then  we  have  in  the  two  cases 
a  vector  (3  such  that  respectively  <p(3  =  a,  or  <p(3  =  0. 
Hence,  if  <pp  =  7,  we  must  have  in  the  two  cases 

7  =  x\  +  yot,        or        7  =  x\. 

Otherwise  the  equation  is  impossible.     Hence 

mip  =  x\  +  za  +  yj3  =  7  +  ua  -f  2/0, 

where  ^>/3  =  a,  <pa  =  0,  or  where  <pfi  =  0  =  ^ck- 

If  ^>  satisfies  the  cubic 

and  no  reduced  equation,  there  are  three  vectors  (of  which 
fi  and  7  are  not  unique)  such  that  <py  =  fi,  <fP  =  a,  <^a  =  0, 
and  then  <pp  =  X,  we  must  have  X  =  xa  +  yft  where  p  is 
any  vector  of  the  form 

p  =  za  +  iCjS  +  1/7. 

If  <p2  =  0,  and  no  lower  degree  vanishes,  then 

<p(x(3  +  2/7)  =  <*j         ^a  =  0,         and        X  =  ua. 

If  <p  =  0,  there  is  no  solution  except  for  <pp  =  0,  where  p 
may  be  any  vector. 

7.  Zero  Roots.  It  is  evident  that  if  one  root  is  zero, 
then  the  region  <p\  where  X  is  any  vector  will  give  us  the 
other  roots.     For  instance  let  <pp  =  Vep.     Then  if  /x  =  Veh, 

cpp,  =  Xe2  —  eSe\,         <p2fi  =  e2/x, 

and  the  other  two  roots  are  ±  V  —  1  •  Te. 

If  two  roots  are  zero,  then  <p2  on  any  vector  will  give  the 
invariant    region    of    the    other    root.     For    instance,  let 


THE   LINEAR   VECTOR   FUNCTION  231 

<pp  =  aSfiyp,  then  aSfiyaSfiyp  =  <p2p.  Hence  cpa  =  aSapy 
gives  the  other  root  as  Sapy  and  its  invariant  line  a. 

In  case  a  root  is  not  zero,  but  is  g\,  if  it  is  of  multiplicity 
one,  then  <p  —  gi  operating  upon  any  vector  will  give  the 
region  of  the  other  root,  or  roots.  If  it  is  of  multiplicity 
two,  then  we  use  (<p  —  g{)2  on  any  vector. 

8.  Transverse.  We  define  now  a  linear  vector  operator 
related  to  <p,  and  sometimes  equal  to  <p,  which  we  shall 
indicate  by  <p' ',  and  call  the  conjugate  of  <p,  or  transverse  of 
<p,  and  define  by  the  equation 

S\<pijl  =  Sn<p'\    for  all    X,  /z. 

For  example,  if  <pp  =  Vap(3,  then  S\<pp  =  S\ap(3  =  SpfiXa, 
and     <p'  =  VPQa  =  <p,     if     <pp  =  Vep,     <p'p  =  —  Vep;    if 
<pp  =  aSfip,  <p'p  =  (3Sap. 
If  a  is  an  invariant  line  of  <p,  (pa  —  ga,  then  for  every  /S 

8p<pa  =  gSaP  =  Sa<p'P, 
or 

Satf  -  g)P  =  0, 

that  is  a  is  perpendicular  to  the  region  not  annulled  by 
<p[  —  g,  that  is  invariant  for  <p'  —  g.  If  we  consider  that 
from  the  definition  we  have  equally 

S\(p2p,  =  Sfxcp'  X,        S\(pziJL  =  Sup'  X, 

it  is  clear  that  <p  and  <p'  have  the  same  characteristic  equa- 
tion and  the  same  general  equation.  They  can  differ  only 
in  their  invariant  regions  if  at  all.  If  then  the  roots  are  all 
distinct,  it  is  evident  that  the  invariant  line  a  of  <p,  is  normal 
to  the  two  invariant  lines  of  <p'  corresponding  to  the  other 
two  roots,  hence  each  invariant  line  of  <p  is  normal  to  the 
two  of  <p'  corresponding  to  the  other  roots,  and  conversely. 
If  now  the  characteristic  equation  is  the  general  equation, 


232  VECTOR  CALCULUS 

so  that  each  function  satisfies  only  the  general  equation, 
let  there  be  two  equal  roots,  g,  whose  shear  region  gives 

<pa=  ga  +  ft         <p@  =  g(3,        let         <py  =  giy. 

Then 

&Vp  =  gSap  +  Sj3p,      S/Vp  =  gSfip,      Sycp'p  =  0i#yp, 
Sapy<p'p  =  g(V(3ySap  +  VyaSfo)  +  F)M0P 

+  giVa(3Syp. 

Therefore  corresponding  to  the  root  g\,  <p'  has  the  in- 
variant line  Vafi,  and  to  the  root  g,  the  invariant  line  V(3y. 
Further  (pf  converts  Vya  into  gVya  +  Vfiy. 

Hence  the  invariant  line  of  g\  for  <p'  is  normal  to  the 
shear  region  of  g,  and  the  shear  region  of  g  for  <pf  is  normal 
to  the  invariant  line  of  g\  for  <p,  but  the  invariant  line  of 
g  for  ip'  is  normal  further  to  the  shear  direction  of  g  for  <p, 
and  the  shear  direction  of  <p'  for  g  is  normal  to  the  invariant 
line  of  (p  for  g. 

In  case  there  are  three  equal  roots,  and  no  reduced  equa- 
tion, we  have 

<pa  =  ga  +  ft         <p/3  =  gfi  +  7,         <£>7  =  PY, 

so  that 

&Vp  =  gSap  +  Sft>,        Wp  =  gS(3p  +  S7p, 

#7<p'p  =  gSyp, 

Sapy<p'p  =  p^Sa/fy  +  VfhfSfo  +  F7CKS7P. 

Hence,  the  invariant  line  of  <p'  is  Vfiy,  its  first  shear  line 
Vya,  and  second  shear  line  Vafi. 

In  case  there  is  a  reduced  equation  with  two  distinct 
roots,  we  have 

<p(xa  -f-  y&)  =  5f(ara  +  yfi),         <P7  =  0i7, 

Sa<p'p  =  gSap,        S/Vp  =  gSfip,        Sy<p'p  =  giSyp, 

Sa&y-<p'p  =  gVfiySap  +  gVyaSfip  +  giVa(3Syp, 


THE   LINEAR  VECTOR   FUNCTION  233 

Hence,  the  invariant  line  of  <p'  corresponding  to  gi  is  normal 
to  the  invariant  plane  of  g  for  <p,  corresponding  to  g  there 
is  an  invariant  plane  normal  to  the  invariant  line  of  gi  for  <p. 
Every  line  in  the  plane  through  the  origin  is  invariant. 
In  case  the  reduced  equation  has  two  equal  roots,  then 

<pa  =  ga  +  ft       <pP  =  gP,       <py  =  gy, 

Sa<p'p  =  gSap  +  S(3p,       Sy<p'p  =  gSyp,       S(3<p'p  =  gSfip, 

Sa(3y<p'p  =  gp  +  Sfa-iVfa), 

Corresponding  to  g,  we  have  then  two  invariant  lines: 
Va.fi,  which  is  perpendicular  to  the  shear  plane  of  <p;  V(3y, 
which  is  perpendicular  to  the  non-shear  region  of  g  and  to 
the  shear  direction  of  g;  also  the  shear  direction  of  <p'  is 
Vfiy,  so  that  the  shear  region  of  <p'  is  determined  by  Vya 
and  Vfiy,  and  is  therefore  perpendicular  to  y. 
The  three  forms  of  <p'  are 

I.     <p'  =  [giVfySaO  +  toVyaSpQ  +  g3Va(3Sy01ISapy; 

II.     <p'  =  faVpySaQ  +  giVyaSpQ  +  aVfaSPQ 

+  g2VaPSyQ]/Safrr, 

III.     <p'  m  g  +  [aVfiySPQ  +  bVyaSyQ  +  cVpySyQ]/SaPy. 

We  may  summarize  these  results  in  the  theorem : 
The  invariant  regions  of  ip'  corresponding  to  the  distinct 
roots  are  normal  to  the  corresponding  regions  of  the  other 
roots  for  <p.  In  case  there  are  repeated  roots,  if  there  is  a 
plane  every  line  of  which  through  the  origin  is  invariant, 
then  every  line  of  the  corresponding  plane  will  also  be  in- 
variant, but  if  there  is  a  plane  with  an  invariant  line  and 
a  shear  direction  in  it,  the  first  invariant  line  of  the  con- 
jugate will  be  perpendicular  to  the  shear  direction  and  to 
the  second  invariant  line  of  <p,  and  the  shear  direction  of  the 
conjugate  will  be  perpendicular  to  the  invariant  lines  of  ip; 

16 


234  VECTOR  CALCULUS 

while  finally,  if  there  is  an  invariant  line,  a  first  shear  direc- 
tion, and  a  second  shear  direction,  then  the  invariant  line 
of  the  conjugate  mil  be  perpendicular  to  the  invariant  line 
and  the  first  shear  direction  of  <p,  the  first  shear  direction 
will  be  perpendicular  to  the  invariant  line  and  the  second 
shear  direction  of  <p,  and  the  second  shear  direction  will  be 
perpendicular  to  the  two  shear  directions  of  <p.  Let  a,  /3,  y 
define  the  various  directions  a  =  V(3y/Sa(3y,  /?  =  Vya/Sa/3y, 
y  =  VaP/Sa(3y,  then  we  have 

<p  =  gioSoi  +  gtffS0  +  gzySy) 
<p'  =  giaSa  +  g2@S(3  +.  g3ySy  J 


or 


or 


(giaSa  +  gJISp  +  afiSa  +  g2ySy) 
\giaSa  +  gi(3S(3  +  aaS(3  +  g2ySy\ 


ig+(aJl+cy)Sa+byS(3 
\g+aaSp+  (b(3  +  ca)Sy . 

9.  Self  Transverse.  It  is  evident  now  that  <p  =  <p'  only 
when  there  are  no  shear  regions,  if  we  limit  ourselves  to 
real  vectors,  and  further  the  invariant  lines  must  be  per- 
pendicular or  if  two  are  not  perpendicular,  then  every 
vector  in  their  plane  must  be  an  invariant,  and  even  in  this 
case  the  invariants  may  be  taken  perpendicular.  Hence 
every  real  self-transverse  linear  vector  operator  may  be 
reduced  to  the  form 

<pp  =  —  aSapgi  —  (3S(3pg2  —  ySypg3, 

where  a  /3  y  form  a  trirectangular  system,  and  where  the 
roots  g  may  be  equal. 

Conversely,  when  <p  —  <p',  the  roots  are  real,  provided 
that  we  have  only  real  vectors  in  the  system,  for  if  a  root 
has  the  form  g  +  ih,  where  i  —  V  —  1,  then  if  the  invariant 


THE   LINEAR   VECTOR   FUNCTION  235 

line  for  this  root  be  X  +  ip,,  where  X  and  p  are  real,  we  have 
<p(\  +  in)  =  (g  +  ih)(k  +  ifi)  =  g\  —  hp  +  i(h\  +  gp) 

=    <p\-\-  iipjJL. 

Therefore 

<p\  =  g\  —  hp,         <pp  =  hX  +  gfx, 
and 

$/*<pX  =  gS\p  —  hp2  =  S\<pp  =  AX2  +  <7$Xju. 

Thus  we  must  have 

^X2  +  hf?  =  0. 

It  follows  that  h  =  0. 

Of  course  the  roots  may  be  real  without  <p  being  self- 
transverse. 

An  important  theorem  is  that  <p  tp'  and  <p'<p  are  self- 
transverse.     For 

Sp<p(p'(T  =  Sa<p<p'p,        Sp<p'<p<r  —  S(T(pf(pp. 

EXERCISE 

Find  expressions  for  <p<p'  and  <p'<p  in  terms  of  a,  /3,  7,  a,  jS,  7. 

10.  Chi  of  p.  We  define  now  two  very  important  func- 
tions related  to  <p  and  always  derivable  from  it.     First 

X*  =  mi  —  <P> 
so  that 

Sa(3y-x<pP  —  pSafi<py  +  pS(3y<pa  +  pSyapfi  —  (paSfiyp 

—  cpfiSyap  —  <pySa(3p 
=  VVaPV<pyp  +  •  •  • 
=  aSp((3<py  —  y<p(3)  +  •  •  •. 

The  other  function  is  indicated  by  \j/^  or  by  xvv  and  defined 

4rp  =  m2  —  mi(p  +  <p2  =  ra2  —  <pxv, 
Sapy-i/z^p  =  pSonpfiipy  +  •  •  •  —  <paSp((3(py  —  y<p(3) 
=  aSp<p(3(py  -\~  f3Sp<py<pa  +  ySp<pa<p(3. 


236  VECTOR   CALCULUS 

We  have  at  once  from  these  formulae  the  following  im- 
portant forms  for  FX/x, 

X„FX/x  =  [aSVlniVpvy  -  Vy<pp)  •  •  .]/SaPy 

=  [aS(V<p'\n  -  V\<p'n)V0y  +  •  •  -]/SaPy 
=  *W  +  V\i/>% 

Whence  we  have  also 

<pV\p  =  miV\ix  —  V\<p'n  —  Vp'XfjL, 
1^FX/x  =  [aSV\»V<pp<py  H ]/Sapy 

=    V<p'\<p'lL. 

Since  it  s  evident  that 

X+  =  x/j        and        \pv,  =  #/, 
we  have  at  once 

x\V\ii  =  V<p\»  +  V\<pfi 

^FX/x  =  V(p\<pji. 

The  two  expressions  on  the  right  are  thus  shown  to  be 
functions  of  FX/x. 

It  is  evident  that  as  multipliers  of  p 

™<i  =  <P  +  X  =  f'  +  X'i 
^2  -  *>X  +  lA  =  *>'x'  +  ^', 

m3  =  ^  =  ^V- 

EXERCISES 

1.  If  <p  =  aiSPiQ  +  a2SM)  +  «*Sfo(),  show  that 

<p'  =  faScHQ  +  02Sa2()  +01&I.O, 
X  =  27/9i7ai(), 
*  =  -  2F/31/825Fala2(), 
mi  =  2*Sau9i,  ra2  =  —  Z£Faia2F(8i/32,  ra3  =  —  Saia2azS0ifi203, 

X'  =  UFaiV/SiO, 
^  =  -  HVaiatSVPiPiQ. 

2.  Show  that  the  irrotational  invariants  of  x  and  ^  are  mi(x)  =  2mh 
m2(x)  =  »ii2  +  m2,  w3(x)  =  Wim2  —  m3;  rai(^)  =  ra2,  m2(^)  =  raira3, 
Wj(^)   =  m32. 


THE   LINEAR   VECTOR   FUNCTION  237 

3.  For  any  linear  vector  function  <p,  and  its  powers  <p2,  <p3,  •  •  • ,  we  have 

mi(<f?)  =  Wi2  —  2ra2,         m2(<p2)  =  m22  —  2raira3,         w3(^)  =  ra32. 
mi(<p3)  =  mi3  —  3wiW2  +  3m3,      m2(<p3)  =  3raira2ra3  —  m23  —  3ra32, 

m3(<p3)  =  m33. 
mi(^4)  =  mi3  —  4mi2w2  +  2w22  +  4raim3 
w2(<p4)  =  w24  —  4wiw22w3  +  2wi2m32  +  4ra2ra32,  m3(^>4)  =  ra34. 

4.  Show  that  for  the  function  <?  +  c,  where  c  is  a  scalar  multiplier, 

mi(<p  +  c)  =  wi(^)  +  3c,        m2((p  +  c)  =  0ts(?)  +  2mi(<p)c  -f  3c2, 
wi3(¥>  +  c)  =  ws(«p)  +  cw2(<p)  +  c2mi(<p)  +  c3. 

5.  Study  functions  of  the  form  x\p  +  ?/x  +  2. 

6.  <p'V<p\<pfi  =  m3V\n;  <p'(V\<pn  —  F/x^X)  =  m2F\M  —  V<p\<pn. 

7.  ^(a^>)  =  aHiv)',     tifilPi)  =  ^(<Pi)-^('Pi)' 

8.  «A(a)  =  a2,     ^[7a()J  =  -  aSaQ,     *(-  0Sa)  =  0. 

+{—  QxiSi  -  gsjSj  -  g3kSk)  =  -  g2g3iSi  —  g3gijSj  -  gig2kSk. 

=  -  VptfiSVaiyi  -  VPtppSVata,  -  VptfiSVa&n. 

10.  For  any  two  operators  <p,  9, 

mi(<pd)  =  mi(M,        m.2(<pd)  =  m2(6<p),        m3(<pd)  =  tr»»(0*). 
mi((p6)  =  mi(<p)mi(0)  +  ra2(<p)  +  ra2(0)  —  m2{6  +  <p). 
m2(<pd)  =  m2{6)m2(<p)  +  m3{<p)-mv{d)  +  ra3(0)-rai(y?) 

-  mtffto  +  *'(*)]. 
m3(<pd)  =  m3(<p)-m3{6). 
rrii(<p  +  0)  =  mi{<p)  +  Wi(0). 

m2(v?  +  0)  =  m2(v?)  +  m2(0)  +  mi(6)'ini(ip)  -  nii(<pO). 
mt(<p  +  0)  =  m3(*>)  +  m3(0)  +  mi[*V(4)  +  0V(*>)]. 

11.  x  can  have  the  three  forms : 

t     (ff*  +  9t)*Sa  +  (g3  +  0i)0S0  +  (oi  +  g2)ySj; 
II.     fo  +  o2)a£5  +  fo  +  fln)/aSg  +  2^x7^7  +  apSa; 
III.     2g  -  (a/3  +  c7)>S£  -  bySfT 

The  operator  x  is  the  rotor  dyadic  of  Jaumann. 

12.  The  forms  of  \f/  for  the  three  types  are 

I.     g2ggaSa  +  g3gi&Sl3  +  gig2ySy; 
II.     gig2aSZ  +  g2gi0S8  +  0i27#t"  -  agtfSZ; 
III.     o2  -  [O03  +  (ab  -  gc)y]Sa  -  bgySfi. 


238  VECTOR  CALCULUS 

13.  An  operator  called  the  deviator  is  defined  by  Schouten,*  and  is 
for  the  three  forms  as  follows: 

I.     (l9i  -  9*  -  gs)aSZ  +  (Itfi  -  0*  -  gi)0S8  +  (lg*  -  gx  -  g%)ySy'; 
II.     (-  fci  -  <7*)(«S5  +  fiSfi)  +  (§0»  -  2^)7^  +  apSZ; 
III.     (o£  +  Cy)Sa  +  bySd. 

It  is  V<p  =  <p  —  S<p,  where  S(<p)  =  \m\. 

14.  Show  that  if  F  (X,  M)  -  -  F  (m,  X)  then 

F  (X,  M)  -  C  (X,  m)Q.VXm, 
where  C  is  symmetric  in  X,  n  and  Q  is  a  quaternion  function  of  VX/i. 
11.  We  derive  from  <p  and  ^'  the  two  functions 

That  there  is  a  vector  e  satisfying  this  last  equation,  and 
which   is   invariant,    is    easily   shown.     For   if   we   form 

™>z(<P  —  <p')>  we  find  that 

S(<p  -  <p')\{<p  -  <p')n(<p  -  (p')v 

=  S(p\<piJL<pi>  —  2$  (p\<p' )jl<p' v  —  ^LSipkcpyup'v—Sip'Xv'mp'v 
=  S\iAi>(m3  —  ra3  +  Wi(^»  —  mi(^')). 

But  it  is  easy  to  see  that  this  expression  vanishes  identically, 
for  the  first  two  terms  cancel,  and  if  <plt  <p2  are  any  two  linear 
vector  functions,  we  have 

=  Siiv<p\kSiiv<p<Lh  +  SjjLvcpiiiSvXip^K  +  SiiV(pivS\mp2K 

+  Sl>\<Pi\SlJlV<p2lJL  +  Sv\<PillSv\(p2lJL  +  Sp\<PipS\/JL(P2H 
+  S\fJL<Pi\SlJLV<p2V  +  S\lJL<PilJ,ST<P2V  +   $XjU<pi  J/jSAjU^ 

=  S^knv  -  mi(<p2<pi) . 

Hence  we  may  under  mi  permute  cyclically  the  vector 
functions.  Again  after  this  has  been  done  we  may  take 
the  conjugate.  Hence  the  expression  above  vanishes,  and 
there  is  a  zero  root  in  all  cases  for  <p  —  <p'.  Further  we 
may  always  write 

*  Grundlagen  der  Vector-  und  Affinor-Analysis,  p.  G4. 


THE   LINEAR  VECTOR   FUNCTION  239 

S\fXV<pp  =    (pXSfJLPp  +    •  •  * 

S\jJLl>'<p'p  =  VjivS\<pfp  +  •  •  • 
=  VfivSipKp  + 
Hence  we  have 

S\nv(<p  -  <p')P  =  VPV(Vfip)cp\  + .... 

From  this  we  have  2eS\pv  =  V(p\Vpv  +  •  •  •  for  every 
noncoplanar  X,  p,  v. 

The  function  <p0  is  evidently  self-transverse,  and  the 
conjugate  of  VeQ  is  —  VeQ.     It  is  easy  to  show  that 

2<peS\pv  =  —  V\V<pp(pv  —  •  •  •. 

The  expressions  Te,  T<pe,  and  Sepe  are  scalar  invariants 
of  <p,  and  these  three  may  be  called  the  rotational  invariants. 
In  terms  of  them  and  the  other  three  scalar  invariants  all 
scalar  invariants  of  <p  or  <p'  may  be  expressed. 

If  there  are  three  distinct  roots,  g\,  g2,  g3,  and  the  corre- 
sponding invariant  unit  vectors  are  yh  y2,  73,  we  may  set 
these  for  X,  p,  v,  and  thus 

2e&7iy2Y3  =  giVjiVy2y3  +  g2Vy2Vyzyi  +  gzVyzVy^y2 
=  (92  —  g3)yiSy2y3  +  (g3  -  gi)y2Sysy2 

+  (gi  —  g2)yzSyiy2. 
2<peSyiy2y3  =  -  g2g3VyxVy2yz  —  g3g\Vy2Vyzyx 

—  gig2Vy3Vyiy2. 

In  case  two  roots  are  equal  and  (pa  =  gxa  +  (3h2, 
<p(3  =  #i/3,  (py  =  g2y,  we  have 

2eSa(3y  =  (g2  -  gi)VyVafi  +  VQVPyh. 

In  case  three  roots  are  equal,  <pa  =  ga-\-h(3,  <pfi  =  gr/5+  ly, 
<py  =  gy 

2eSa(3y  =  h(3V(3y  +  lyVya. 

It  is  evident,  therefore,  that  if  the  roots  are  distinct  and 


240  VECTOR  CALCULUS 

the  axes  perpendicular  two  and  two,  that  «  =  0;  if  two 
roots  are  equal  and  the  invariant  line  of  the  other  root  is 
perpendicular  to  the  plane  of  the  equal  roots,  then  it  is  the 
direction  of  e;  and  if  the  three  roots  are  equal,  and  if  the 
invariant  line  is  perpendicular  to  the  two  shear  directions, 
then  €  is  in  the  plane  of  the  invariant  line  and  the  second 
shear. 

12.  Vanishing  Invariants.  The  vanishing  of  the  scalar 
invariants  of  (p  leads  to  some  interesting  theorems. 

If  Wi  =  0,  there  is  an  infinite  set  of  trihedrals  which  are 
transformed  by  <p  into  trihedrals  whose  edges  are  in  the 
faces  of  the  original  trihedral.  If  ^transforms  any  trihedral 
in  this  manner,  mi  =  0,  and  there  is  an  infinite  set  of  trihe- 
drals so  transformed. 

We  choose  X,  n,  v  for  the  edges  of  the  vertices,  and  if  <p\ 
is  coplanar  with  /z,  7,  <pix  with  v,  X,  and  <pv  with  X,  ju,  the 
invariant  mi  =  0.  If  mi  =  0,  we  choose  X,  ju,  arbitrarily, 
and  determine  v  from  ScpXnv  =  0  =  SXcpuv.  Then  also 
S\n<pp  =  0. 

The  invariant  m2  vanishes  if  <p  transforms  a  trihedral 
into  another  whose  faces  pass  through  the  edges  of  the  first. 
The  converse  holds  for  any  infinity  of  trihedrals. 

EXERCISES 

1.  Show  that  if  a,  fi,  7  form  a  trirectangular  system 

mi  =  —  Sa<pa  —  S/3<pl3  —  Sy<py 

and  is  invariant  for  all  trirectangular  systems, 

m2(<p<p')  =  T*<poc  +  TV/?  +  T*<py, 
TV(X)  =  S2\<pa  -f  £2Xv/3  +  S2\<py. 

2.  Study  the  functions  for  the  ellipsoid  and  the  two  hyperboloids 

-  <p  =  a^aSa  ±  b~2fiSl3  ±  c^ySy. 
3    Study  the  functions 

ZmVaVQct,  <P  +  VaVQa,  a^VoapQ, 

r-VpVaQ,         V -vVaQ.fi. 


THE   LINEAR   VECTOR   FUNCTION 


241 


4.  Show  that 


V  <pp  =  2e  —  mi, 
\/Sp<pp  =  —  2  (pop, 


VAp  =  —  2<pt  —  m2, 
\7Vp<pp  =  2Sep  +  Wip  —  3<pp, 


wherein  <p  is  a  constant  function.  Hence  (pop  may  always  be  repre- 
sented as  a  gradient  of  a  scalar,  Sep  as  a  convergence  of  a  vector,  and 
m,\p  —  3<pp  (deviation)  as  a  curl.  We  may  consider  also  that  Wi  is  a 
convergence  and  e  is  a  curl,  ra2  a  convergence  and  <pe  a  curl. 

5.  An  orthogonal  function  is  defined  to  be  one  such  that 

ip<p'  =  1. 

Show  that  an  orthogonal  function  can  be  reduced  to  the  form 

ip  =  ()  cos  0  -  sin  0-70/3  =  (lT  cos  9)0800  =  0**l*Q0-*l* 

or  —  /3(0/"-)+1()/?-(fl./7r)-1  which  is  a  rotation  about  the  axis  /3  through 
the  angle  —  0,  or  such  a  rotation  followed  by  reflection  in  the  plane 
normal  to  /?. 

6.  Study  the  operator  <p112. 

7.  Show  that 


m 

■i(<po)  =  mh 

m2 

(<Po)  = 

mi  +  e2,         m3(<p0)  =  m3  ■ 

f  5-6V56. 

Hence  if 
Tf 

Te 

=  0, 

m2(<p)  =  m2(<p0). 

u 

8. 

Show  that 

Se<pt 

i  =  0, 

m3(<p)  =  m3(<p0). 

mAVe{)] 

=  o, 

m*[VeQ]  =  TV,        m3[Ve()]  = 

0. 

9. 

Show  that 

e(x)  =  - 

■    6, 

«(x) 

-  "  *e'         e(^_1)  =  "  m3 

ipe. 

10, 
11, 

,  Show  that 

.  if  0  =  y./M), 

\p(<Po) 

=  *0  +  aSe(). 

rni(d)  = 

2S/3e, 

m-i 

(0)  =  -  S/3*>/3,         m3(d)  = 

o, 

"2(a2  —  aSa), 


12.  If  *>  =  F-a(), 

,p2»  = 

^2n+l    =  a2n7a()> 

13.  For  any  two  operators  <p,  0, 

2eM)   =  2e(^o0o)  +  X(<p)e(6)  +  x{0)e(<p)  +  V-e(<p)e(6). 


242  VECTOR  CALCULUS 

In  particular 

14.  An  operator  ^>  is  a  similitude  when  for  every  unit  vector  a, 
T^a  =  c,  a  constant. 

Show  that  the  necessary  and  sufficient  condition  is 

<p'<p  =  c2. 

Any  linear  transformation  which  preserves  all  angles  is  a  similitude. 

15.  If  <p  =  aSi  +  0Sj  +  ySk,  then  <p'  =  iSa  +jSp  +  kSy,  and 
^j^'  =  —  ctSa  —  fiSfi  —  7$7, 

mi(*V)  =  Pa  +  P/3  +  7*7,       m,(^^')  =  PFa/S  +  7*7/37  +  T^a, 
mz(<p<p')  =  —  S2a0y. 

13.  Derivative  Dyadic.  There  is  a  dyadic  related  to  a 
variable  vector  field  of  great  importance  which  we  will 
study  next.  It  is  called  the  derivative  dyadic,  since  it  is 
somewhat  of  the  nature  of  a  derivative,  as  well  as  of  the 
nature  of  a  dyadic.  This  linear  vector  function  for  the 
field  of  a  will  be  indicated  by  Da  and  defined  by  the  equation 

D.=  ~  SQV-<r. 

It  is  evident  at  once  that  if  we  operate  upon  dp,  we  arrive 
at  da.  This  function  is,  therefore,  the  operator  which  en- 
ables us  to  convert  the  various  infinitesimal  displacements 
in  the  field  into  the  corresponding  infinitesimal  changes 
in  the  field  itself. 
The  expression 

SdpDJp  =  Cdf, 

where  C  is  a  constant  and  dt  a  constant  differential,  repre- 
sents an  infinitesimal  quadric  surface,  the  normals  at  the 
ends  of  the  infinitesimal  vectors  dp  being  Dadp. 

Let  us  consider  now  the  field  of  a,  containing  the  con- 
gruence of  vector  lines  of  <r.  Consider  a  small  volume 
given  by  8p  at  the  point  whose  vector  is  p,  and  let  us  sup- 


THE   LINEAR  VECTOR   FUNCTION  243 

pose  it  has  been  moved  to  a  neighboring  position  given  by 
the  vector  lines  of  the  congruence,  that  is,  p  becomes 
p  +  adt.     Then  p  +  8p  becomes 

p+8p  +  dt(<r  +  DM, 

that  is  to  say,  dp  has  become 

(1  +  Dadt)8p. 

Hence  any  area  V8\p82p  becomes,  to  terms  of  the  first 
order  only, 

V8lP82p  +  dt(V8lPD,82p  +  VDJlP82P). 

The  rate  of  change  with  regard  to  t  of  the  vector  area 
V8ip82p  is  therefore 

X  (Dff)V8lP82p. 

Likewise,  the  infinitesimal  volume  S8ip82p8sp  is  trans- 
formed into  the  volume 

S8ip82p83p  +  dt{S8ip82pDa8zp  +  S8ipDa82p83p 

+  SDa8ip82p83p). 

The  rate  of  increase  of  the  volume  is,  therefore,  miS8ip82p83p. 
In  other  words  if  we  displace  any  portion  of  the  space  of 
the  medium  so  that  its  points  travel  infinitesimal  distances 
along  the  lines  of  the  congruence  of  a,  by  amounts  propor- 
tional to  the  intensity  of  the  field  at  the  various  points,  then 
the  change  in  any  infinitesimal  line  in  the  portion  of  space 
moved  is  given  by  dtDff8p,  the  change  in  any  infinitesimal 
area  is  given  by  x'(Da)dt- Area,  and  the  change  in  an 
infinitesimal  volume  is  midt  times  the  volume. 

In  case  a  defines  a  velocity  field  the  changes  mentioned 
will  actually  take  place.  We  have  here  evidently  a  most 
important  operator  for  the  study  of  hydrodynamics.  If 
adt  is  the  field  of  an  infinitesimal  strain,  then  Da8p  is  the 


244  VECTOR  CALCULUS 

displacement  of  the  point  at  dp.  Evidently  the  operator 
plays  an  important  part  in  the  theory  of  strain,  and  con- 
sequently of  stress.  Further,  (we  shall  not  stop  to  prove 
the  result  as  we  do  not  develop  it)  for  any  vector  a  a 
function  of  p  we  have  an  expansion  analogous  to  Taylor's 
theorem,  in  the  series 

h2 
<r(p0  +  ha)  =  (r(po)  +  hD^  +  -^  (-  &*V)Z).a 

+  |  (SaV)2Daa  +  •••• 

This  formula  is  the  basis  of  the  study  of  the  singularities 
of  the  congruence.  For  if  cr(p0)  =  0,  then  the  formula  will 
start  with  the  second  term,  and  the  character  of  the  con- 
gruence will  depend  upon  the  roots  of  Dff.  In  brief  the 
results  of  the  investigation  of  Poincare  referred  to  above 
(p.  38)  show  that  if  none  of  the  roots  is  zero,  we  have  the 
cases : 

1.  Roots  real  and  same  sign,  the  singularity  is  a  node. 

2.  Roots  real  but  not  all  of  the  same  sign,  a  faux. 

3.  One  real  root  of  same  sign  as  real  part  of  other  two, 
a  focus. 

4.  One  real  root  of  sign  opposite  the  real  part  of  others, 
a  faux-focus. 

5.  One  real  root,  other  two  pure  imaginaries,  a  center. 
If  one  or  more  roots  vanish,  we  have  special  cases  to  con- 
sider. 

The  invariants  of  Da  are  easily  found,  and  are 

mi  =  —  SV<r,     e  =  ^Vxja,     m2  =  —  %SVViV2V<tkt2, 
D*e  =  iV-VViV2V<ri<T2,     m3  =  |SViV2V3<W-20-3. 

After  differentiation,  the  subscripts  are  all  removed.  The 
related  functions  are 


THE   LINEAR   VECTOR   FUNCTION  245 

BJ  =  -  v&r(),     X  =  -  VVV*Q,     %'  =  ~  V-VQV-*, 

$  =  -  jrviV2^K72(),     y  =  -  i&oviVs-Wi*!. 

In  a  strain  a  the  dilatation  's  rab  the  density  of  rotation 
(spin)  is  e,  and  in  other  cases  we  can  interpret  m\  and  e  in 
terms  of  the  convergence  and  the  curl  of  the  field.  In 
case  a  is  a  field  of  magnetic  induction  due  to  extraneous 
causes,  and  a  is  the  unit  normal  of  an  infinitesimal  circuit 
of  electricity,  then  %'«  is  the  negative  of  the  force  density 
per  unit  current  on  the  circuit.  In  any  case  we  might  call 
—  x'V8iP82p  the  force  density  per  unit  circuit.  Since  x' 
is  not  usually  self-transverse,  the  force  on  circuit  a  has  a 
component  in  the  direction  jS  different  from  the  component 
in  the  direction  a  of  the  force  on  circuit  ft. 

Recurring  to  Stokes'  and  Green's  theorems  we  see  that 

fdpa  =  ff  -  WVdlPd2P'(T 

=  2ffS8lP82pe  -Sfx'V8lP82P. 

It  is  clear  that  the  circulation  in  the  field  of  a  is  always 
zero  unless  for  some  points  inside  the  circuit  e  is  not  zero. 
The  torque  of  the  field  on  the  circuit  vanishes  for  any 
normal  which  is  a  zero  axis  of  x'«  From  these  it  is  clear  that, 
if  we  have  a  linear  function  <pdP,  in  order  that  it  be  an  exact 
differential  da  we  must  have  the  necessary  and  sufficient 
conditions 

VVvO  m  0. 

For  if  tf<pdP  =  0,  then  <pVUvV  =  0  for  all  Uv,  whence 
the  condition.     The  converse  is  easy. 

The  invariant  m3  in  the  case  of  the  points  at  which  a  =  0 
will  be  sometimes  positive,  sometimes  negative.  A  theorem 
given  originally  by  Kronecker  enables  us  to  find  what  the 
excess  of  the  number  of  roots  at  which  ra3  is  positive  over 
the  number  of  roots  at  which  ra3  is  negative  is.*     We  set 

*  Picard,  Traite  d'Analyse,  Vol.  I,  p.  139. 


246  VECTOR  CALCULUS 

t  =  fa/To*        and        7  =  -  J-  ffSdvr] 

47T 

then  the  integral  will  vanish  for  any  space  containing  no 
roots,  and  will  be  the  excess  in  question  for  any  other  space. 
We  could  sometimes  use  this  theorem  to  determine  the 
number  of  singularities  in  a  region  of  space  and  something 
about  their  character.     It  is  evident  that  <SVr  =  0. 

The  operator  (Dc)0  =  \(D0  +  DJ)  is  called  the  deforma- 
tion of  the  field,  and  the  operator  Ve()  the  rotation  of  the 
field. 

In  case  a  is  a  unit  vector  everywhere,  then  DJa  =  0, 
and  since  the  transverse  has  a  zero  root,  Da  itself  must  have 
a  zero  root.  There  is  one  direction  then  for  which  Dffa  =  0. 
The  vector  lines  given  by  Vadp  =  0  are  the  isogons  of  the 
field.  In  case  there  are  two  zero  roots  the  isogons  are  any 
lines  on  certain  isogon  surfaces. 

EXERCISES 

1.  Study  the  fields  given  by 

a  =   —  p,  a  =  Up/p2,  a  =  Vap,  a  =  aSfip,  a  =  Vap/p3. 

2.  Show  that  if  a  is  a  function  of  p, 

a  +  da  =   —  V  o[Spo<r  —  %Spo<ppo]  —  \V Po^V <* 
=  VVoihVvpo  -  Wpo<PPo]  -  lSV<r, 

where  Vo  operates  only  on  p0,  and  <p  =  —  <rSS7  0-  The  first  form 
expresses  a  +  da  as  a  gradient  and  a  term  dependent  on  the  curl  of  a, 
the  second  as  a  curl  and  a  term  dependent  on  the  convergence  of  a. 
po  is  an  infinitesimal  vector. 

3.  If  a  =  FVr,  Da  =  ZV. 

14.  Dyadic  Field.  If  <p  is  a  linear  vector  operator  de- 
pendent upon  p,  we  say  that  <p  defines  a  dyadic  field.  For 
every  point  in  space  there  will  be  a  value  of  <p.  Since  there 
is  always  one  root  at  least  for  <p  which  is  real,  with  an  in- 
variant line,  there  will  be  for  every  point  in  space  a  direction 


THE   LINEAR  VECTOR   FUNCTION  247 

and  a  numerical  value  of  the  root  which  gives  the  real 
invariant  direction  and  root.  These  will  define  a  con- 
gruence of  lines  and  a  numerical  value  along  the  lines. 
In  case  the  other  axes  are  also  real,  and  the  roots  are  distinct 
or  practically  distinct,  there  will  be  two  other  related  con- 
gruences. The  study  of  the  structure  of  a  dyadic  field 
from  this  point  of  view  will  not  be  entered  into  here,  but 
it  is  evidently  of  considerable  importance. 

EXERCISES 

1.  If  <p  =  uQ,  then  the  gradient  of  the  field  is  Vw.  The  vorticity 
of  the  field  is  VV  <p()  —  VVuQ.  The  gradient  in  any  case  is  v'V, 
a  vector. 

2.  If  <p  =  VaQ,  the  gradient  is  —  V\7<r,  the  vorticity  is 

QSS7*+DV  m  -  x{D,r). 

3.  If  <p  =  <tSt(),  the  gradient  is  aSVr  —  DT<r,  the  vorticity  is 
WvStQ  +  V<rDy().     The  gradient  of  the  transverse  field  is  tS\7<t 

-  DaT,  the  vorticity  VX/tSvQ  +  VtD<j{). 

4.  If  ip  =  VadQ,  the  gradient  is  -  70(V)<r  +  VadV,  the  vor- 
ticity is 

S\7<T-d()  +S*V'-0'Q -*S$[(V i)Q  +£<r0(). 

For  the  transverse  field  we  have 

the  gradient  is  —  0'FV'o-  —  0VV<r, 
the  vorticity  7v  W«r()  +  W'dVa'Q. 

5.  If  <p  =  D,r  the  gradient  of  the  field  is  —  VV,  the  concentration  of 
<r,  and  the  vorticity  is  D  vva  •     The  gradient  of  the  transverse  field  is 

—  V^Vo",  while  the  vorticity  is  zero. 

6.  If  <p  =  VV0(),  the  gradient  is  FV0V,  where  both  V's  act  on  0, 
and  the  vorticity  is  V20()  -  V&V0(). 

7.  If  <p  =  De(a),  the  gradient  is  —  7^,  the  vorticity  is  Dvve<r- 

8.  If  f>  =  to,  the  gradient  is  2e(07V0). 

9.  For  any  <p 

Vm1  =  <PV  +2e  (FWO  ), 
Vm2  =  2  €  {<pW<p'  +  FW'  0  I, 
Vw3  =  2  e  [V  (Vi  M  *' 
-t  W  Vx]. 


248  VECTOR  CALCULUS 

15.  The  Differentiator.  We  define  the  operator  —  SQ  V 
as  the  differentiator,  and  indicate  it  by  D.  It  may  be  used 
upon  quaternions,  vectors,  scalars,  or  dyadics. 

As  examples  we  have,  D  being  the  transverse 

Bv„  =  VaDr()  -  VtD0Q,        DSar  -  SQD.r  +  S()DTa, 
DVaa  =  -  VaD.Q,  DmiM  =  mriDJ, 

DeM=  e(DJ,        Dv=  -S()V  •*>(). 

16.  Change  of  Variable.  Let  F  be  a  function  of  p,  and 
p  a  function  of  three  parameters  u,  v,  w.     Let 

A  =  ad/du  +  f3d/dv  +  yd/dw, 

where  a,  /3,  y  form  a  right-handed  system  of  unit  vectors. 
Then  we  have  the  following  formulae  to  pass  from  expres- 
sions in  terms  of  p  to  differential  expressions  in  terms  of  the 
parameters. 

AF  =  -  AiSPlVFt 
FA' A"  =  |FAi'A2"£Fpip2Fv'V", 
SA'A"A'"  -  -  i<SAi'A2"A8,"iSpiP2PsSV' V'V". 

As  instances 

-  SVv=  A'VV'V, 
VA<r=  VV"T(r"A'. 

Notations 
Dyadic  products 
4>(a),  <f>'(a),  <f)Va(  ),  Va(f>(  ),  Hamilton,  Tait,  Joly,  Shaw. 
<l>'af  a-4>,  <j)  X  a,  aX  <j>,  Gibbs,  Wilson,  Jaumann,  Jung. 

Reciprocal  dyadic 
4>~l,   Hamilton,   Tait,   Joly,   Gibbs,   Wilson,   Burali-Forti, 

Marcolongo,  Shaw. 
q~l,  Timerding. 
I6I"1,  filie. 


THE   LINEAR   VECTOR   FUNCTION  249 

The  adjunct  dyadic 
\j/  =  m(f)'~l,  Hamilton,  Tait,  Joly,  Shaw. 
WO2,  Gibbs,  Wilson,  Macfarlane. 
R{a),  Burali-Forti,  Marcolongo. 
x((f>,  (f>),  Shaw. 
D4>~1,  Jaumann,  Jung. 

The  transverse  or  conjugate  dyadic 
<f>',  Hamilton,  Tait,  Joly. 
0,  Taber,  Shaw. 

<f>c,  Gibbs,  Wilson,  Jaumann,  Jung,  Macfarlane. 
K(ct),  Burali-Forti,  Marcolongo. 
\b  /  ,  Elie. 

The  planar  dyadic 
X  =  Wi  —  (f>r,  Hamilton,  Tait,  Joly. 
4>J  —  <f>c,  Gibbs,  Wilson. 
—  </>/,  Jaumann,  Jung. 
CK(a),  Burali-Forti,  Marcolongo. 
x(0),  Shaw. 

Self-transverse  or  symmetric  part  of  dyadic 
<f>ot  Hamilton,  Tait,  Shaw. 
$,  Joly. 

<f>f,  Gibbs,  Wilson. 
[</>],  Jaumann,  Jung. 
D(a),  Burali-Forti,  Marcolongo. 
\  b  /,  Elie. 
\  b°  / ,  Elie.     In  this  case  expressed  in  terms  of  the  axes. 

Skew  part  of  dyadic 
\{4>  —  </>')  =  V-e(  ),  Hamilton,  Tait,  Joly,  Shaw. 
</>",  Gibbs,  Wilson. 
II,  Jaumann,  Jung. 
Va  A ,  Burali-Forti,  Marcolongo. 

17  i 


250  VECTOR  CALCULUS 

\  b  / ,  £lie. 

Sin  <f>,  Macfarlane. 

Mixed  functions  of  dyadic 
X«>,  0),  Shaw. 
\<f>l  0,  Gibbs,  Wilson. 
R{(f>,  0),  Burali-Forti,  Marcolongo. 

Vector  of  dyadic 
e,  Hamilton,  Tait,  Joly. 
<£x,  Gibbs,  Wilson. 
(f>r8,  —  <}>/,  Jaumann,  Jung. 
Va,  Burali-Forti,  Marcolongo. 
E,  Carvallo. 
R  =  Te,  filie. 
c(<£),  Shaw. 

Negative  vector  of  adjunct  dyadic 
<f>e,  Hamilton,  Tait,  Joly. 
0-0x,  Gibbs,  Wilson. 
<t>-<f>r8,  Jaumann,  Jung. 
olVol,  Burali-Forti,  Marcolongo. 
«x(</>>  <f>)>  Shaw. 

Square  of  pure  strain  factor  of  dyadic 
4><f>',  Hamilton,  Tait,  Joly. 
</></>c,  Gibbs,  Wilson. 
{(f)}2,  Jaumann,  Jung. 
aKa,  Burali-Forti,  Marcolongo. 
[6],  filie. 
</></>',  Shaw. 

Dyadic  function  of  negative  vector  of  adjunct 
<f>2e,  Hamilton,  Tait,  Joly,  Shaw. 
<f>2-4>x,  Wilson,  Gibbs. 


THE   LINEAR   VECTOR   FUNCTION  251 

02  -0/,  Jaumann,  Jung. 

a2Va,  Burali-Forti,  Marcolongo. 

K2,  Elie. 

Scalar   invariants   of  dyadic.     Coefficients   of  characteristic 

equation 
m" ',  ra',  m,  Hamilton,  Tait,  Joly,  Carvallo. 
1%,  h,  h,  Burali-Forti,  Marcolongo,  Elie. 
F,  G,  H,  Timerding. 
0S,  (</>2)s,  03,  Gibbs,  Wilson, 
mi,  ra2,  ra3,  Shaw. 

fc,  ] 

4>8*,         >■  •  •  •  03,  Jaumann,  Jung. 

-  w,  J 

cos  </>•••  03,  Macfarlane. 

(Mer  scalar  invariants 
™>i(<f>o2),  mi(00'),  2(rai2  —  m2),  rai(00')> 

wi[x(0,  *)>  0L  Shaw. 
[08]2«,  {0js2>  [01/,  •'*  -j  Jaumann,  Jung. 
•  •  •,  •  •  •,  •  • .,  0  :  0,  0*  0  :  ft  Gibbs,  Wilson. 
Elie  uses  ifi  for  $e0e. 

Notations  for  Derivatives  of  Dyadic 
In  these  V  operates  on  0  unless  the  subscript  n  indicates 
otherwise. 

Gradient  of  dyadic 
V0,  Tait,  Joly,  Shaw. 

Dyadic  of  gradient.     Specific  force  of  field 
0V,  Tait,  Joly,  Shaw, 
grad  a,  Burali-Forti,  Marcolongo. 

-3 — ,  Fischer. 
dr 


252  VECTOR  CALCULUS 

Transverse  dyadic  of  gradient 
0'V,  Tait,  Joly. 
grad  Ka,  Burali-Forti,  Marcolongo. 

—r^-y  Fischer. 

V  -<t>,  Jaumann,  Jung. 

Divergence  of  dyadic 

-  SV<f>( ),  Tait,  Joly,  Shaw. 

X  grad  Ka,  Burali-Forti,  Marcolongo. 

Vortex  of  dyadic 
VV4>( ),  Tait,  Joly,  Shaw. 
Rot  a,  Burali-Forti. 

V  X  0,  Jaumann,  Jung. 

Directional  derivatives  of  dyadic 

-  S( )  V  •  0.    Sa'1  V  ■  <l>a.    ScT1  V  -<t>Va(),  Tait,  Joly,  Shaw. 
S(a,  (  )),  Burali-Forti. 

P  ,  IX*,  Fischer. 
da  da 

Burali-Forti,  Marcolongo. 


(»<>)<»• 


Gradient  of  bilinear  function 
ju„(Vn,  «),  Tait,  Joly,  Shaw. 
<£(/z)a,  Burali-Forti. 

Bilinear  gradient  function 
ju(Vn,  un),  Tait,  Joly,  Shaw. 
\//(n,  u),  Burali-Forti. 

Planar  derivative  of  dyadic 
<f>nVVn(  ),  Tait,  Joly,  Shaw. 

X-^>  Fischer, 


CHAPTER  X 
DEFORMABLE  BODIES 

Strain 

1.  When  a  body  has  its  points  displaced  so  that  if  the 
vector  to  a  point  P  is  p,  we  must  express  the  vector  to  the 
new  position  of  P,  say  P',  by  some  function  of  p,  cpp, 
then  we  say  that  the  body  has  been  strained.  We  do  not 
at  first  need  to  consider  the  path  of  transition  of  P  to  P'. 
If  cp  is  a  linear  vector  function,  then  we  say  that  the  strain 
is  a  linear  homogeneous  strain.  We  have  to  put  a  few 
restrictions  upon  the  generality  of  <p,  since  not  every  linear 
vector  function  can  represent  a  strain.  In  the  first  place 
we  notice  that  solid  angles  must  not  be  turned  into  their 
symmetric  angles,  so  that  SipKcpyupvlSKp.v  must  be  positive, 
that  is,  ra3  is  positive.  Hence  (p  must  have  either  one  or 
three  positive  real  roots.  The  corresponding  invariant  lines 
are,  therefore,  not  reversed  in  direction. 

2.  When  <p  is  self-conjugate  there  are  three  real  roots 
and  three  directions  which  form  a  trirectangular  system. 
The  strain  in  this  case  is  called  a  pure  strain.  Any  linear 
vector  function  can  be  written  in  the  form 

#rr  V.{**9."fI0«  =  p-1V(<pV)()-p, 

where 

q-i()q  =  (wT'V 

The  function  <p<p'  is  self -conjugate  and,  therefore,  has  three 
real  roots  and  its  invariant  lines  perpendicular.  If  we  set 
7r  =  V  ((p<p')y  then  7r2  =  ipip'.  Let  the  cubic  in  <p<p'  be 
G3  -  MXG2  +  M2G  -  M3  =  0.  Then  from  the  values 
given  in  Chapter   IX,  p.   237,   for  the  coefficients  of   <p2 

253 


254  VECTOR   CALCULUS 

in  terms  of  those  of  <p  we  have  (the  coefficients  of  the  cubic 
in  w  being  pu  p2,  p3) 

Mi  =  pi*  -  2p2f        M2  =  p22  —  2pm,        M3  =  pz2, 
whence  we  have 

P!4  -  2(Mi  +  8M3)pi2  -  \m2MzVl  +  MS  -  4M22M3  =  0. 

Thence  we  have  pi,  p2,  and  p3. 

Now  if  the  invariant  lines  of  <p<p'  are  the  trirectangular 
unit  vectors  a,  0, 7,  we  may  collect  the  terms  of  <p  in  the  form 

<P  =  aaSa'Q  +  bpSP'Q  +  cySy'Q, 

where  a,  b,  c  are  the  roots  of  V  <p<p'  =  w  and  a! ,  fi',  7'  are 
to  be  determined.     Hence  <p'  =  aa'SaQ  +  •  •  •  and 

-  tp'ip  =  tfa'Sa'Q  +  VP'Sp'Q  +  WO- 

But  also 

^'  =  _  otefo  _  fc0S0  -  c27#7, 

since  a,  /?,  7  are  axes  of  <p(p',  and  a2,  b2,  c2  are  roots.  Now 
we  have 

<p'a  =  —  act',         <p'/3  =  —  b(3',         <p'y  =  —  cy', 

hence 

<p(pra  =  a2a  =  —  a2otSa'ct'  —  ab(3Sa'(3'  —  acySot'y'. 

Thus  we  have  a:'2  =  —  1,  Set' (3'  =  0  =  Sa'y',  and  similar 
equations,  so  that  a',  (3',  y'  are  unit  vectors  forming  a  tri- 
rectangular system,  and  indeed  are  the  invariant  lines  of 
<p'<p.     We  may  now  write  at  once 

7r  =  —  aaSct  —  b/3S(3  —  cySy, 
q~\)q  =  -  aSa'  -  fiSfi*  -  ySy'. 

This  operator  obviously  rotates  the  system  a',  (3',  y'  into 


DEFORM  ABLE   BODIES  255 

the  system  a,  (3,  y,  as  a  rigid  body.  That  the  function  is 
orthogonal  is  obvious  at  a  glance,  since  if  we  multiply  it 
by  its  conjugate  we  have  for  the  product 

-  aSa  -  PSP  -  ySy  =  1(). 

Reducing  it  to  the  standard  form  of  example  five,  Chapter 
IX,  p.  236,  we  find  that  the  axis  is  UV(aa'  +  (3(3'  +  77')  and 
the  sine  of  the  angle  of  rotation  \TV{aa!  +  $8'  +  77') • 

EXAMPLES 

(1).  Let  <p  =  VeQ-     Then 

<p'  =  -  VeQ,         <P<P'  =  ~  VeVeQ  =  eSe()  -  e2. 

The  axes  are  e  for  the  root  0,  and  any  two  vectors  a,  j8 
perpendicular  to  e,  and  these  must  be  taken  so  that  a(3  =  Ue, 
the  roots  that  are  equal  being  T2e.     We  may  therefore  write 

<p  =  TeaS(3  -  Te(3Sa  =  V-eQ, 

which  was  obvious  anyhow.  Hence  we  have  for  q~lQq  the 
operator 

aSp-0Sa=  V(VaP)Q, 

and  this  is  a  rotation  of  90°  about  Va(3  =  Ve  of  90°.  The 
effect  of 

7T  =   Te(-  aSa  -  (3S(3) 

is  to  give  the  projection  of  the  rotated  vector  on  the  plane 
perpendicular  to  e,  times  Te.  That  is,  finally,  VeQ  rotates 
p  about  €  as  an  axis  through  90°  and  annuls  the  component 
of  the  new  vector  which  is  parallel  to  e. 

(2).  Consider  the  operator  g  —  aS(3Q  where  a,  jS  are  any 
vectors.  It  is  to  be  noticed  that  we  must  select  of  all  the 
square  roots  of  (p<pf  that  one  which  has  its  roots  all  positive. 
It  is  obvious  that  j)  =  q. 


256  VECTOR  CALCULUS 

3.  The  strain  converts  the  sphere  Tp=  r  into  the  ellip- 
soid 7V-1p  =  r,  or 

WW  =  -  r\ 

This  is  called  the  strain  ellipsoid.  Its  axes  are  in  the  direc- 
tions of  the  perpendicular  system  of  (p<p'  —  tt2.  The  ellip- 
soid Sp<p'<pp  =  —  r2  is  converted  into  the  sphere  Tp  =  r. 
This  is  the  reciprocal  strain  ellipsoid.  Its  axes  are  in  the 
directions  of  the  principal  axes  of  the  strain.  The  exten- 
sions of  lines  drawn  in  these  directions  in  the  state  before 
the  strain  are  stationary,  and  one  of  them  is  thus  the  maxi- 
mum, one  the  minimum  extension. 

4.  A  shear  is  represented  by 

<PP  —  P  ~  fiSap, 

where  Sa/3  =  0.  The  displacement  is  parallel  to  the  vector 
/3  and  proportional  to  its  distance  from  the  plane  Sap  =  0. 
There  is  no  change  in  volume  since  ms  =  1. 

If  there  is  a  uniform  dilatation  and  a  shear  the  function  is 

<pp  =  gp-  fiSap. 

The  change  in  volume  is  now  g3.  The  equation  is  easily 
seen  to  be 

(<P  -  9?  =  0. 

This  is  the  necessary  and  sufficient  condition  of  a  dilatation 
and  a  shear,  but  this  equation  alone  will  not  give  the  axes 
and  the  shear  plane,  of  course. 

5.  The  function  <pp  =  gqpcT1  ~  qfiq~lSap  is  a  form  into 
which  the  most  general  strain  can  be  put  which  is  due  to 
shifting  in  a  fixed  direction,  U(5,  planes  parallel  to  the  fixed 
plane  Sap  =  0  by  an  amount  proportional  to  the  perpen- 
dicular distance  from  the  fixed  plane,  then  altering  all 
lines  in  the  ratio  g}  and  superposing  a  rotation.     This  is 


DEFORMABLE   BODIES  257 

any  strain.     We  simply  have  to  put  <p'<p  into  the  form 

<p'<p  =  b2  +  X£ju  +  ftSk, 
where 

S\fx  =  i(a2  +  c2-  2b2),         T\n  =  K«2  -  c2), 
and  then  we  take 

g=b,        a  -  -  X,         bp  =  n-  IXrK*  ~  c)2- 

The  rotation  is  determined  as  before. 

6.  All  the  lines  in  the  original  body  that  are  lengthened 
in  the  same  ratio,  say  g,  are  parallel  to  the  edges  of  the  cone 
TcpUp  =  g  or  SUp(<pf<p  —  g2)Up  =  0,  or  in  terms  of  X,  /z, 
2SX  UpSfx  Up  =  b2  -  g2,  sin  u-smv=  (b2-  g2)/(a2  -  c2) , 
where  u  and  v  are  the  angles  the  line  makes  with  the  cyclic 
planes  of  the  cone  Scpp<pp  =  —  T2p. 

7.  The  displacement  of  the  extremity  of  p  is 

5  =  a  —  p  =  O  —  l)p, 

which  can  be  resolved  along  p  and  perpendicular  to  p  into 
the  parts 

p(/Sp~Vp  —  1)  +  pVp~l<pp. 

The  coefficient  of  p  in  the  first  term  is  called  the  elongation. 
It  is  numerically  the  reciprocal  of  the  square  of  the  radius 
of  the  elongation  quadric: 

Sp(<p0  —  l)p  =  —  1, 

the  radius  being  parallel  to  p. 

The  other  component  may  be  written  Vep  +  Vcpopp~l-  p, 
where  e  is  the  invariant  vector  of  <p,  the  spin-vector. 

8.  If  now  the  strain  is  not  homogeneous,  we  must  con- 
sider it  in  its  infinitesimal  character.  In  this  case  we  have 
again  the  formula  da  =  —  SdpV  -tr  =  cpdp,  where  a  is  now 
the  displacement  of  P,  whose  vector  is  p,  and  a  +  da  that  of 


258  VECTOR  CALCULUS 

p  -f-  dp,  provided  that  we  can  neglect  terms  of  the  second 
order.     If  these  have  to  be  considered, 

da  =  -  SdpV  a  +  i(SdpV)  VV 
=  (pdp  —  %SdpV  -  (pdp. 

We  may  analyze  the  strain  in  the  case  of  first  order  into 

<P  =  (fo  +  VeQ. 

Since  now  €  =  \V\7<r,  if  e  =  0,  it  follows  that  a  =  VP 
and  there  is  a  displacement  potential  and 

p '«  -  VSVP(). 

The  strain  is  in  this  case  a  pure  strain.  If  e  is  not  zero, 
there  is  rotation,  about  e  as  an  axis,  of  amount  Te.  In  any 
case  the  function  <p0  determines  the  changes  of  length  of  all 
lines  in  the  body,  the  extension  e  of  the  short  line  in  the 
direction  Up  being 

—  SUppoUp. 

The  six  coefficients  of  <p0,  of  form  —  Sa<po(3,  where  a,  ft 
are  any  two  of  the  three  trirectangular  vectors  a,  ft  7, 
are  called  the  components  of  strain.  Three  are  extensions 
and  three  are  shears,  an  unsymmetrical  division. 

9.  In  the  case  of  small  strains  the  volume  increase  is 
—  S\7<7,  and  this  is  called  the  cubic  dilatation.  If  it 
vanishes,  the  strain  takes  place  with  no  change  of  volume, 
that  is,  with  no  change  of  density.  A  strain  of  this  char- 
acter is  called  a  transversal  strain.  There  is  a  vector 
potential  from  which  a  can  be  derived  by  the  formula 

a  =  VVt,         SVt  =  0. 

There  is  no  scalar  potential  since  we  do  not  generally  have 
also  VVo-  =  0.     Indeed  we  have 

2e  =  VV<r  =  WVVr  =  V2r  -  VSVr  =  V2r. 


DEFORMABLE   BODIES  259 

This  would  give  us  the  integral 

\t  =  \irfffejr-dv. 

The  integration  is  over  the  entire  body. 

This  strain  is  called  transverse  because  in  case  we  have  a 
a  function  of  a  single  projection  of  p,  on  a  given  line,  say  a, 
so  that 

a  =  afvx  +  /3f2-x  +  yf3-x, 
SVo-  =  —  /i  =  0,        fi  =  constant, 

and  all  points  are  moved  in  this  direction  like  those  of  a 
rigid  body.  We  may  therefore  take  the  constant  equal  to 
zero,  and  /i  =  0,  so  that 

Saa  =  0. 

Hence  every  displacement  is  perpendicular  to  the  line  a. 

10.  When  V\/a  =  0,  we  call  the  strain  longitudinal;  for, 
giving  <j  the  same  expression  as  in  §  9,  we  see  that  we  have 

Wa  =0  =  7/2'  -  fifs',         and        /2  =  0  =  /3, 
Vaa  =  0. 

Hence  we  have  all  the  strain  parallel  to  a. 

11.  In  case  the  cubical  dilatation  iSVo"  =  0,  the  strain 
is  purely  of  a  shearing  character,  and  if  the  curl  VVv  =  0, 
the  strain  is  purely  of  a  dilatational  character.  Since  any 
vector  a  can  be  separated  into  a  solenoidaJ  and  a  lamellar 
part  in  an  infinity  of  ways,  it  is  always  possible  to  separate 
the  strain  into  two  parts,  one  of  dilatation  only,  the  other 
of  shear  only. 

If  we  write  a  =  VP  +  V\/t,  then  we  can  find  P  and  r 
in  one  way  from  the  integrals 

P  =  lir.ffSS<T'VTp-W, 
r  =  -  \TT'fffVa'VTp-l-dvf,         p  =  p'  -  Pc. 


260  VECTOR  CALCULUS 

The  integrations  extend  throughout  the  body  displaced. 
This  method  of  resolution  is  not  always  successful,  and 
other  formulae  must  be  used.  (Duhem,  Jour,  des  Math., 
1900.) 

12.  The  components  are  not  functionally  independent, 
but  are  subject  to  a  set  of  relations  due  to  Saint  Venant. 
These  relations  are  obvious  in  the  quaternion  form,  equiva- 
lent to  six  scalar  equations.     The  equation  is 

V-V<PoVV()  =  0,        if        <p=SQV-<r, 

where  both  V's  operate  on  <p0.  The  equation  is,  further- 
more, the  necessary  and  sufficient  condition  that  any  linear 
vector  function  <p  can  represent  a  strain.  The  problem  of 
finding  the  vector  a  when  <p  is  a  given  linear  and  vector 
function  of  p  consists  in  inverting  the  equation 

<p  =  —  S()  V  -cr.     (Kirchkoff,  Mechanik,  Vorlesung  27.) 

It  is  evident  that  if  we  operate  upon  dp,  we  have 

<pdp  =  do. 

Hence  the  problem  reduces  to  the  integration  of  a  set  of 
differential  equations  of  the  ordinary  type. 

EXAMPLES 

(1).  If  (p  =  VeQ,  we  have  or  =  Vep.  Prove  Saint  Venant's 
equations. 

(2).  If  <p~  p-lV{)p-\  then  a  =  Up.  Prove  Saint  Ven- 
ant's equations. 

13.  In  general  when  we  do  not  have  small  strains,  we 
must  modify  the.  preceding  theory  somewhat.  The  dis- 
placement will  change  the  differential  element  dp  into 

dpi  =  dp  —  SdpV-<r. 

The  strain  is  characterized  when  we  know  the  ratio  of  the 
two  differential  elements  and  this  we  may  find  by  squaring 


DEFORMABLE   BODIES  261 

so  as  to  arrive  at  the  tensor 

(dpi)*  =  Sdp[l  -  2vSa  +  V'S(r'(T"SV"]dp- 

The  function  in  the  brackets  is  the  general  strain  function, 
which  we  will  represent  by  <£.  It  is  easily  clear  that  if 
<p  =  —  SQV'<r  then 

*  =  (1  +  <p)(l  +  <p')  =  (1  +  *>)(1  +  <p)'. 

Of  course  $  is  self-conjugate.  Its  components  Sa&fi  are 
also  called  components  of  strain.  If  <p  is  infinitesimal,  we 
may  substitute  (1  +  2<po)  for  <£. 

The  cubical  dilatation  is  now  found  by  subtracting  1  from 

SdipidhpidtPi/Sdipdtpdtp  =  m3(l  +  <p)  =  1  +  A. 

Evidently  (1  +  A)2  =  m3($).  The  alteration  in  the 
angle  of  two  elements  is  found  from 

-  suq.  +  <p)\u(i<p)y. 

If  angles  are  not  altered  between  the  infinitesimal  elements, 
the  transformation  is  conformal,  or  isogonal.     In  such  case 

Eti&k'  =  s2\ys\$\s\'$y. 

For  example,  if  <p  =  VaQ, 

sua  +  <p)\Q.  +  <p)v  =  sxx', 

when  Sa\  =  0  =  Sa\'. 

14.  This  part  of  the  subject  leads  us  into  the  theory  of 
infinitesimal  transformations,  and  is  too  extensive  to  be 
treated  here. 

On  Discontinuities 

15.  If  the  function  <j  is  continuous  throughout  a  body, 
it  may  happen  that  its  convergence  or  its  curl  may  be  dis- 
continuous.    The  consideration  of  such  discontinuities  is 


262  VECTOR  CALCULUS 

usually  given  at  length  in  a  discussion  of  the  potential 
functions.  Here  we  need  only  the  elements  of  the  theory. 
We  make  use  of  the  following  general  theorem  from  analysis. 

Lemma.  If  a  function  is  continuous  on  one  side  of  a  sur- 
face for  all  points  not  actually  on  the  surface  in  question,  and 
if,  as  we  approach  the  surface  by  each  and  every  path  leading 
up  to  a  point  P,  the  gradient  of  the  function,  or  its  directional 
derivatives  approach  one  and  the  same  limit  for  all  the  paths; 
then  the  differential  of  this  function  along  a  path  lying  on  the 
surface  is  also  given  by  the  usual  formula, 

—  SdpV  -q  =  dq,        dp  being  on  the  surface. 

[Hadamard,  Lecons  sur  la  propagation  des  ondes,  etc., 
p.  84,  Painleve,  Ann.  Ecole  Normale,  1887,  Part  1,  ch.  2, 
no.  2.] 

In  the  case  of  a  vector  a  which  has  the  same  value  on 
each  side  of  a  surface,  which  is  the  value  on  the  surface, 
and  is  the  limiting  value  as  the  surface  is  approached,  at 
all  points  of  the  surface,  we  have  on  one  side  of  the  surface 

da  —  —  Sdp\7  •&  =  <pidp. 

On  the  opposite  side 

da  =  —  Sdp\7  -<r  =  <p2dp. 

If  now  these  two  do  not  agree,  but  there  is  a  discontinuity 
in  <p,  so  that  <p2  —  tp\  is  finite  as  the  two  paths  are  made 
to  approach  the  surface,  then  designating  the  fluctuation  or 
saltus  of  a  function  by  the  notation  [],  we  have  in  the  limit 

[da]  =  (<p2  —  <Pi)dp  =  [<p]dp. 

But  since  a  does  not  vary  abruptly,  [da]  along  the  surface 
is  zero,  hence  for  dp  on  the  surface 

[<p]dp  =  0, 


DEFORMABLE   BODIES  263 

and  therefore 

M  =  —  vSv> 

where  v  is  the  unit  normal,  \x  a  given  vector.  That  is  to 
say,  we  have  for  the  transition  of  the  surface 

[S()V-a]  =  »Sv. 
Whence 

[SVcr]  =  Spix, 
[W<t]  =  Vvix. 

These  are  conditions  of  compatibility  of  the  surface  of  dis- 
continuities and  the  discontinuity;  or  identical  conditions, 
under  which  the  discontinuities  can  actually  have  the  sur- 
face for  their  distribution. 

16.  If  *S/x^  =  0,  then  [S  Vo"]  =  0,  and  the  cubic  dilatation 
is  continuous. 

Since  Svvjjl  =  0  =  Sv[V\7<t]  =  [SpV<t],  the  normal  com- 
ponent of  the  curl  of  a  is  continuous,  and  the  discontinuity 
is  confined  to  the  tangential  component.     Likewise 

Sfivn  =  0  =  [S/xVo-], 

and  the  component  along  ijl  is  continuous.  Hence  V\7(r 
can  be  discontinuous  only  normal  to  the  plane  of  /jl,  v. 

17.  In  case  a  itself  is  discontinuous,  the  normal  com- 
ponent of  a  as  it  passes  the  surface  of  discontinuity  cannot 
be  discontinuous  without  tearing  the  surface  in  two.  Hence 
the  discontinuity  is  purely  tangential.  It  can  be  related 
to  the  curl  of  a  as  follows.  .        . 

Consider  a  line  on  the  surface,  of  infinitesimal  length,  and 
an  infinitesimal  rectangle  normal  to  the  surface,  and  let 
the  value  of  a  at  the  two  upper  points  differ  only  infinites- 
imally,  as  likewise  at  the  two  lower  points,  but  the  differ- 
ence at  the  two  right  hand  points  or  at  the  two  left  hand 


264  VECTOR  CALCULUS     . 

points  be  finite,  so  that  a  has  a    discontinuity   in  going 
through  the  surface  equal  to  [a].     Then 

fSbpa  =  ffSK(AWa) 

around  the  rectangle,  when  k  is  normal  to  the  rectangle. 
But  the  four  parts  on  the  left  for  the  four  sides  give  simply 

Sid*}, 

where  8p  is  a  horizontal  side  and  equal  to  V-wTSp.     Hence 
we  have  for  every  k  tangential  to  the  surface 

SkV v[a]  -  Sk  Urn  (AW<r)IT8p. 

Dropping  all  infinitesimals,  we  have 

Vv[(t]  =  Lim  AVVcr/Tdp. 

Tangential  discontinuities  may  therefore  be  considered 
to  be  representable  by  a  limiting  value  of  the  curl  multi- 
plied by  an  infinitesimal  area,  as  if  the  surface  of  discon- 
tinuity were  the  locus  of  the  axial  lines  of  an  infinity  of  small 
rotations  which  enable  one  space  to  roll  upon  the  other. 
The  expression  \[<j]  is  the  strength  of  this  sheet. 

A  strain  is  not  irrotational  unless  such  surfaces  of  dis- 
continuity are  absent.  But  we  have  shown  above  that  a 
continuous  strain  may  imply  certain  surfaces  of  discon- 
tinuity in  its  derivatives  of  some  order.  If  V\7cr  =  0, 
everywhere,  then  Vv[u]  =  0,  and  such  discontinuity  as 
exists  is  parallel  to  v. 

The  derivation  above  applies  to  any  case,  and  we  may 
say  that  if  a  field  is  irrotational,  any  discontinuity  it  pos- 
sesses must  be  normal  to  the  surface  of  discontinuity. 

Integrating  in  the  same  way  over  the  surface  of  a  small 
box,  we  would  have 

ffSv[<r]ds  =  SV<T'V, 


DEFORMABLE   BODIES  265 

where  v  is  the  infinitesimal  volume.     But  this  gives 

Sv[a]  =  vSV  (r/surface. 

If  then  $Vo"  =  0  everywhere,  the  discontinuity  of  a  is 
normal  to  the  normal,  that  is,  it  is  purely  tangential.  These 
theorems  will  be  useful  in  the  study  of  electro-dynamics. 

Kinematics  of  Displacements 
18.  In  the  case  of  a  continuous  displacement  which  takes 
place  in  time  we  have  as  the  vector  a  the  velocity  of  a 
moving  particle,  and  if  p  is  the  vector  from  a  fixed  point 
to  the  particle,  then  dp/dt  =  a.  It  is  necessary  to  distin- 
guish between  the  velocity  of  the  particle  and  the  local 
velocity  of  the  stream  of  particles  as  they  pass  a  given  fixed 
point  in  the  absolute  space  which  is  supposed  to  be  sta- 
tionary. The  latter  is  designated  by  d/dt.  Thus  dcr/dt  is 
the  local  rate  of  change  of  the  velocity  at  a  certain  point. 
While  da/dt  is  the  rate  of  change  of  the  velocity  as  we  follow 
the  particle.  It  is  easy  to  see  that  for  any  quaternion  q 
the  actual  time  rate  of  change  is 

dq/dt  =  dq/dt  —  SaV  -q. 

We  have  thus  the  acceleration 

da/dt  =  da/dt  -  SaV-cr  =  (d/dt  +  <p)a. 

If  the  infinitesimal  vector  dp  is  considered  to  be  displaced, 
we  have 

bdp/dt  =  -  S5pV'(r. 

Since  the  rotation  is  \V\7a  dt,  the  angular  velocity  of  turn 
of  the  particle  to  which  dp  is  attached  is  |FVo".  This  is 
the  vortex  velocity.  Likewise  the  velocity  of  cubic  dilata- 
tion is  —  S\/a. 

The  rate  of  change  of  an  infinitesimal  volume  dv  as  it 
18 


266  VECTOR  CALCULUS 

moves  along  is 

—  SV<T'dv. 

The  equation  of  continuity  is  d(cdv)  =  0,  where  c  is  the 
density,  or 

dc/dt  +  c{-  SV<r)  =  0. 

That  is,  we  have  for  a  medium  of  constant  mass 

dc/dt  =  cSVv- 

That  is,  the  density  at  a  moving  point  has  a  rate  of  change 
per  second  equal  to  the  density  times  the  convergence  of 
the  velocity. 

It  may  also  be  written  easily 

dc/dt  =  SVW. 

This  means  that  at  a  fixed  point  the  velocity  of  increase  in 
density  is  equal  to  the  convergence  of  the  momentum  per 
cubic  centimeter. 

19.  When  FVo"  =  0,  the  motion  is  irrotational,  or  dila- 
tational,  and  we  may  put  a  =  VP,  where  now  P  is  a  veloc- 
ity-potential, which  may  be  monodromic  or  polydromic. 
When  SVcr  =  0,  the  motion  is  solenoidal  or  circuital,  and 
we  may  write  a  =  VVr  where  &Vr  =  0.  r  is  the  vector 
potential  of  velocity.  The  lines  e  =  \V\7<r  become  in  this 
case  the  concentration  of  Jr.  The  lines  of  a  are  the  vortex 
lines  of  r,  and  the  lines  of  e  are  the  vortex  lines  of  a. 

20.  If  a  is  continuous,  and  the  equation  of  a  surface  of 
discontinuity  of  the  gradient  dyadic  of  a  and  of  a'  is  /  =  0, 
where  now  a  is  a  displacement  and  a'  is  da/dt  the  velocity, 
we  have  certain  conditions  of  kinematic  compatibility. 
These  were  given  by  Christoffel  in  1877-8  and  are  found  as 
follows.     We  have 

M  =  o,       [_0ov«W<-jtfi> 


DEFORMABLE   BODIES  267 

in  the  case  in  which  the  time  t  is  not  involved;  and  for  a 
moving  surface  in  which  /  is  a  function  of  t  as  well  as  of  p, 
we  would  have 

[-SOV-<r]=-»SUvfO, 

["  S Tt  V<T]  =  "  mS i  Uvf=  Mf  /*V/=M=-Gm. 

This  gives  us  the  discontinuity  in  the  time  rate  of  change  of 
the  displacement  of  a  point  as  it  passes  from  one  side  to  the 
other  of  the  moving  surface.  The  equation  of  the  surface 
as  it  moves  being  /(/>,  t),  we  have  in  the  normal  direction 

-  SdpV-f+dtf  =  0, 

that  is,  since  dp  is  now  Uvfdn,  dn/dt  —  —  f'/T\/f  =  G, 
where  / '  is  the  derivative  of  /  as  to  t  alone.  In  words, 
at  any  point  on  the  instantaneous  position  of  the  moving 
surface  the  rate  of  outward  motion  of  the  point  of  the 
surface  coinciding  with  the  fixed  point  in  space  is 
G  =  —f'jTS/f.  The  moving  surface  of  discontinuity  is 
called  a  wave  and  G  the  rate  of  propagation  of  the  wave  at 
the  given  point.  We  may  now  read  the  condition  of  com- 
patibility above  in  these  words:  the  abrupt  change  in  the 
displacement  velocity  is  given  by  a  definite  vector  p.  at 
each  point  multiplied  by  the  negative  rate  of  propagation 
of  the  wave  of  displacement,  that  is,  if  G  is  the  rate  of 
propagation, 

[o-'j  =  -  Gp,        and        [SVff]  =  -  SpVvf  =  -  S/iv. 

21.  The  preceding  theorem  becomes  general  for  discon- 
tinuities of  any  order  in  the  following  way.  Let  the  func- 
tion a  and  all  its  derivatives  be  continuous  down  to  the 
(n  —  l)th,  then  we  can  write 

[SQiV'SQzV  —  S0*-iV-*]«0, 


268  VECTOR  CALCULUS 

whence,  differentiating  along  the  surface  of  discontinuity 
as  before,  we  find  in  precisely  the  same  manner 

•  [S()iV  •  •  •  SO. V  •*)  =  nSOiUvfSihUvf  •  •  •  SQnUvf, 

since  at  a  given  point  on  the  fixed  surface  V/  is  constant. 
And  if  we  insert  dp/dt  in  m  parentheses  (m  <  ri),  we 
shall  have,  since  the  surface  is  moving, 

=  -  »G»SQiUvf  •  •  •  S0n-mUvf(-l)m. 

In  particular  for  m  —  2  =  n,  we  have 

W)  =  mG2, 

which  is  the  discontinuity  in  the  acceleration  of  the  dis- 
placement. 

If  m  =  1,  n  =  2, 

[SOW]  =  -  nGSQUVf. 

From  this  we  derive  easily 

[SVff'l  =  -  GS»Uvf=  -  GSfxp. 
[W<r']  =  -  GVfxUvf=  ~  GVfip. 

22.  The  nth.  derivatives  of  Saa  are 

[S()iV  •  •  •  SQnV-Saa]  =  SQiUvf  •  •  ■  SQnUVfSap. 

If  then  we  hold  the  surface  fixed  and  consider  a  certain 
point,  the  discontinuity  in  the  nth  derivative  of  the  ratio 
of  two  values  of  the  infinitesimal  volume  which  has  two 
perpendicular  directions  on  the  surface  and  the  third  along  . 
the  normal  will  be  given  by  the  formula 

SQiUvf  ■  ■  ■  SOnllVfSnUvf. 
In  case  we  have  a  material  substance  that  has  mass  and 


DEFORMABLE   BODIES  269 

density  and  of  which  the  mass  remains  fixed,  we  have 

c/cq  =  volo/voi, 
log  c  —  log  Co  =  log  v0  —  log  V, 
V  log  c  =  —  V  log  v/v0  =  —  Vo/v-V(clvo). 

Therefore  from  the  formula  above  we  have  since  v0/v  =  1 
in  the  limit 

[SOiV  •  •  •  S()nV  log  c]  =  SQiUVf  •  •  •  SQnUvfSfjiUvf. 

In  particular  for  the  case  of  discontinuities  of  order  two- 
we  have 

[Vlogc]=  UvfSfiUvf. 

23.  These  theorems  may  be  extended  to  the  case  in  which 
the  medium  is  in  motion  as  well  as  the  wave  of  discontinuity. 

Stress 

24.  In  any  body  the  stress  at  a  given  point  is  given  as  a 
tension  or  a  pressure  which  is  exerted  from  some  source 
across  an  infinitesimal  area  situated  at  the  point.  The 
stress  real  y  consists  of  two  opposing  actions,  being  taken 
as  positive  if  a  tension,  negative  if  a  pressure.  It  is  as- 
sumed that  the  stress  taken  all  over  the  surface  of  an 
infinitesimal  closed  solid  in  the  body  will  be  a  system  of 
forces  in  equilibrium,  to  terms  of  the  first  order.  This  is 
equivalent  to  assuming  that  the  stress  on  any  infinitesimal 
portion  of  the  surface  is  a  linear  function  of  the  normal, 
that  is 

6  =  ZVv. 

25.  We  have  therefore  for  any  infinitesimal  portion  of 
space  inside  the  body 

ffQdA  =  ffZdv  =  0. 

But  by  Green's  theorem  this  is  equal  to  the  integral  through 


270  VECTOR  CALCULUS 

the  infinitesimal  space  J J VHV  =  0.     Hence  SV  =  0. 
In  this  equation  S  is  a  function  of  p,  and  V  differentiates  S. 

26.  In  case  the  portion  of  space  integrated  over  or 
through  is  not  infinitesimal,  this  equation  (in  which  S  is 
no  longer  a  constant  function)  remains  true  if  there  is 
equilibrium;  and  if  there  are  external  forces  that  produce 
equilibrium,  say  £  per  unit  volume,  then  the  density  being 
c,  we  have 

SV  +  c£  =  0 
for  every  point. 

In  case  there  is  a  small  motion,  we  have 

EV  +  c£  =  co". 

27.  Returning  to  the  infinitesimal  space  considered,  we 
see  that  the  moment  as  to  the  origin  of  the  stress  on  a 
portion  of  the  boundary  will  be  VpSJJv  and  the  total 
moment  which  must  vanish,  considering  S  as  constant,  is 

ffVpZdv  =  fffVpttdv, 
hence 

FpHv  =  0  =  €(S). 

We  see  therefore  that  S  is  self-conjugate. 

EXAMPLES 

(1).  Purely  normal  stress,  hydrostatic  stress.  In  this  case 
S  is  of  the  form  pS  =  gp,  where  g  is  +  for  tension,  —  for 
pressure,  and  is  a  function  of  p  (scalar,  of  course). 

(2).  Simple  tension  or  pressure. 

H  =  —  paSa. 
(3).  Shearing  stress. 

H  =  -  p(aSp  +  PSa), 
|S  not  parallel  to  a. 


DEFORMABLE   BODIES  271 

(4).  Plane  stress. 

8  -  giaSa  +  g2(3S(3. 
(5).  Maxwell's  electrostatic  stress. 

H=  l/87r-FvP()VP, 

where  P  is  the  potential. 

28.  The  quadric  Spap  =  —  C  is  called  the  stress  quadric. 
Its  principal  axes  give  the  direction  of  the  principal  stresses. 
Since  Sp  is  the  direction  of  the  normal  we  may  arrive  at  a 
graphical  understanding  of  the  stress  by  passing  planes 
through  the  center,  and  to  each  construct  the  conjugate 
diameter.  This  will  give  the  direction  of  the  stress,  and 
since  Tap  is  inversely  proportional  to  the  perpendicular 
from  the  origin  on  the  tangent  plane  at  p,  if  we  lay  off  on 
the  conjugate  diameter  distances  inversely  as  the  per- 
pendiculars, we  shall  have  the  vector  representation  of  the 
stress.  When  the  diameter  is  normal  to  its  conjugate  plane, 
there  will  be  no  component  of  the  corresponding  vector 
that  is  parallel  to  the  plane,  that  is,  no  tangential  stress. 
Such  planes  will  be  the  principal  planes  of  the  stress. 

It  is  evident  that  a  stress  is  completely  known  when  the 
self-conjugate  linear  vector  function  H  is  known,  which 
depends  therefore  upon  six  parameters.  We  shall  speak, 
then,  of  the  stress  H,  since  H  represents  it.  This  proposi- 
tion is  sometimes  stated  as  follows:  stress  is  not  a  vector 
but  a  dyadic  (tensor).  From  this  point  of  view  the  six 
components  of  the  stress  are  taken  as  the  coordinates  of  a 
vector  in  six-dimensional  space.  These  components  in  the 
quaternion  notation  are,  for  a,  (3,  y,  a  trirectangular  system, 

-  SaXa,        -  S/3E0,        -  SyZy,        -  Saafi  =  -  S(3Za, 
-  SpEy  =  -  SyZp,         -  SyZa  =  -  Sa3y. 


272  VECTOR  CALCULUS 

That  is, 

Xx       Y y      Zt,  Xy  =    YXi  Y  t  =   Zy,  Zix  ~   X  2. 

It  is  easy  to  see  now  that  certain  combinations  of  these 
component  stresses  are  invariant.  Thus  we  have  at  once 
the  three  invariants  mi,  m2,  m3,  which  are 

Xx  ~r*  Yy~\~  %zt    YyZz  -f-  ZgXx  ~r  XxYy  —  Yz  —  Zx —  Xy  , 
XtY yZ z  -\-  ZXyY ZZX       XXY z         Y yZx        ZzXy  . 

For  any  three  perpendicular  planes  these  are  invariant. 

EXERCISE 

What  are  the  principal  stresses  and  principal  planes  of  the  five  ex- 
amples given  above? 

29.  Returning  to  the  equation  of  a  small  displacement, 
we  may  write  it 

er"  =  i  +  <TlEV. 

Hence  the  time  rate  of  storage  or  dissipation  of  energy  is 

W'=-  fffSa'Zvdv. 

The  other  terms  of  the  kinetic  energy  are  not  due  to  storage 
of  energy. 

Now  we  have  an  experimental  law  due  to  Hooke  which  in 
its  full  statement  is  to  the  effect  that  the  stress  dyadic  is  a 
linear  function  of  the  strain  dyadic.  The  latter  was  shown 
to  be 

<Po=  -^S()V-<7+  V&rOJ. 

The  law  of  Hooke  then  amounts  to  saying  that  S  is  a  linear 
function  of  a  and  V  where  V  operates  upon  a,  and  owing 
to  the  self-conjugate  character  of  <p,  we  must  be  able  to 
interchange  V  and  a,  that  is, 

S  =  6[(),  V,  a}. 


DEFORMABLE   BODIES  273 

First,  it  follows  that  if  the  strain  <p0  is  multiplied  by  a 
variable  parameter  x,  that  the  stress  will  be  multiplied  by 
the  same  parameter.  We  have  then  for  a  parametric  change 
of  this  kind  which  we  may  suppose  to  take  place  in  a  alone 
a'  =  ax' .  Hence  for  a  gradually  increasing  a,  we  would 
have 

W  =  -  xx'fffSaSVdv, 

w  =  -  iyyy&rEv  &% 

if  x  runs  from  0  to  1.  This  gives  an  expression  for  the 
energy  if  it  is  stored  in  this  special  manner.  If  the  work  is 
a  function  of  the  strain  alone  and  not  dependent  upon  the 
way  in  which  it  is  brought  about,  W  is  called  an  energy- 
function.  It  is  thus  seen  to  be  a  quadratic  function  of  the 
strain.  In  case  there  is  an  energy  function,  we  have  for 
two  strain  functions  due  to  the  displacements  crlf  a2 

Si  =  e[(),  en,  Vi],        H2  =  G[(),  o-2,  v2]- 

The  stored  energy  for  the  two  displacements  must  be  the 
same  either  way  we  arrange  the  displacements,  hence  we 
have 

So-2e3[V3,  *i»  Vi]  =  (Scr1e4[V*i  <r2,  V2], 

where  the  subscripts  3,  4  merely  indicate  upon  what  V  acts. 
This  is  equivalent  to  saying  that  so  far  as  vector  function 
is  concerned,  in  the  form 

SaG[(3,  7,  5] 

we  can  interchange  a,  (3  and  y,  5.  Since  S  is  self-conjugate, 
0  is  self-conjugate,  and  we  can  interchange  a  and  (3.  From 
the  nature  of  the  strain  function  we  can  interchange  y,  8. 
Of  course,  in  the  forms  above  we  cannot  interchange  the 
effect  of  the  differentiations. 


274  VECTOR  CALCULUS 

We  have  in  this  way  arrived  at  six  linear  vector  functions 

<P\l       <f22       <P32       <f23       <fn       <Pl2> 

wherein  we  can  interchange  the  subscripts,  and  where 
<Pn  =  0[Q,a,a]  •••  ^23=  6[(),ft7]    v\, 

a  /3  7  being  a  trirectangular  system  of  unit  vectors.  We 
have  further  a  system  of  thirty-six  constituents  Cmu  cni2, 
•  •  •  where 

Cim  =  —  Sa<pn<x,         C1112  =  —  Sa<pn<x,  •  •  •, 

each  of  the  six  functions  having  six  constituents.  These 
are  the  36  elastic  constants.  If  there  is  an  energy  function, 
they  reduce  in  number  to  only  21,  for  we  must  be  able  to 
interchange  the  first  pair  of  numbers  with  the  last  pair. 
There  are  thus  left 

3  forms  emu  6  of  em%,  3  of  Cim,  3  of  C1212,  3  of  C2311,  3  of  02m. 

In  theories  of  elasticity  based  upon  a  molecular  theory 
and  action  at  a  distance  six  other  relations  are  added  to 
these  reducing  the  number  of  elastic  constants  to  15.  These 
relations  are  equivalent  to  an  interchange  of  the  second 
and  third  subscript  in  each  form,  thus  Cim  =  Ci2is-  These 
are  usually  called  Cauchy's  relations,  but  are  not  commonly 
used.     (See  Love,  Elasticity,  Chap.  III.) 

Remembering  the  strain  function  <p0,  we  can  interpret 
these  coefficients  with  no  difficulty,  for  we  have 

—  SaipoCXj  •  fty, 

the  stress  dyadic  due  to  the  strain  component  —  Sa&oaj, 
where  a;,  a;  are  any  two  of  the  three  a,  (3,  y.  cijki  is  the 
component  of  the  stress  across  a  plane  normal  to  otj  in  the 
direction  at  due  to  the  strain  component  —  Sak<Poai- 


DEFORMABLE    BODIES  275 

EXAMPLES 

(1).  If  Sij  =  —  Soti<pocxj,  show  that  we  have  for  the  energy 
function 

W  =  ^CnnSn  +  2cii22SnS22  +  i^c12i2s122 

+   201223^12^23  +   SCni2*ll*l2  +   2Cii2SSnS23. 

(2).  When  there  is  a  plane  of  symmetry,  say  in  the  direc- 
tion normal  to  7,  all  constants  that  involve  7  an  odd  number 
of  times  vanish,  for  the  solid  is  unchanged  by  reflection  in 
this  plane.  Only  thirteen  remain.  If  there  are  two  per- 
pendicular planes  of  symmetry,  normal  to  (3,  y,  the  only 
constants  left  are  of  the  types 

ClUli       C1122,       Ci212j 

the  plane  normal  to  a  is  thus  a  plane  of  symmetry  also. 
There  are  nine  constants.  This  last  case  is  that  of  tesseral 
crystals. 

(3).  If  the  constants  are  not  altered  by  a  change  of  a  into 
—  a,  (3  into  —  (3,  as  by  rotation  about  7  through  a  straight 
angle,  then  the  plane  normal  to  7  is  a  plane  of  symmetry. 

(4).  Discuss  the  effect  of  rotation  about  7  through  other 
angles. 

(5).  When  the  energy  function  exists  we  have 

0(X,  fi,  v)  -  90*,  X,  v)  =  -  VvQV\\x,      where  6'  =  6. 

30.  A  body  is  said  to  be  isotropic  as  to  elasticity  when  the 
elastic  constants  are  not  dependent  upon  directions  in  the 
body.  In  such  case  the  energy  function  is  invariant  under 
orthogonal  transformation.  It  must,  therefore,  be  a  function 
of  the  three  invariants  of  <po,  i»i,  ra2,  m3.  The  last  is  of 
third  degree,  while  the  energy  function  is  a  quadratic 
and  therefore  can  be  only  of  the  form 

W  =  -  Pmi  +  Am?  +  Bm2. 


276  VECTOR  CALCULUS 

P  is  zero  except  for  gases  and  is  then  positive.  The  con- 
stant A  refers  to  resistance  to  compression,  and  is  positive. 
B  is  a  constant  belonging  to  solids. 

The  form  given  the  quadratic  terms  by  Helmholtz  is 

Amx2  +  Bm2  =  iHml2  +  £C[2mi2  -  6m2]. 

The  []  is  the  sum  of  the  squares  of  the  differences  of  the 
latent  roots  of  <po.  The  constant  H  refers  to  changes  of 
volume  without  change  of  form,  and  in  such  change  it  is  the 
whole  energy,  for  if  there  is  no  change  of  form,  the  roots 
are  all  equal  and  the  other  term  is  zero.  C  refers  to  changes 
of  form  without  change  of  volume,  since  it  vanishes  if  the 
roots  are  equal  and  is  the  whole  energy  if  there  is  no  cubical 
expansion  m\.  For  perfect  fluids  C  =  0. 
The  form  given  by  Kirchoff  is 


Km^tpo2)  +  Kdrm2. 
From  which  we  have 

B-C  =  2KB,     3C  =  2K,     H=  2K(d  +  |),      C  =  \K. 
We  may  write  for  solids,  liquids,  and  gases 

W  =  Rdm?  +  Kmifao*)  -  Pmx. 
Later  notation  gives  2K6  =  X,  K  =  /x,  that  is, 

W  =  |Xmi2  +  iirriiicpo2)  —  Pm\. 

The  constants  X,  \x  are  the  two  independent  constants  of 
isotropic  bodies. 

We  now  have  for  the  stress  function  in  terms  of  the  strain 
function 

S  =  Xrai  -f-  2/i^o. 

EXAMPLES 
(1).  In  the  case  of  a  simple  dilatation  we  know  S  =  p 


DEFORMABLE   BODIES  277 

and  we  have  for  <po 

<Po=  -  JOSOV-ap  +  ASapQ)  =  a(). 

Substituting  in  the  equation  above,  we  have 

()p  m  X(3o)  +  2Mo(). 

The  cubical  dilatation  is  thus 

3a  =  p/(X  +  |m)  =  p/», 

where  A:  is  called  the  modulus  of  cubical  compression. 
(2).  For  a  simple  shear 

<p,  =  -  a/2-[aSPQ  +  g&xOL    ™i  =  0> 
S  =  -  a/z[«<Sj8()  +  0&*()]. 

If  the  tangential  stress  is  T,  then  T  =  a/j,.     M  is  the  shear 
modulus  or  simple  rigidity. 

(3).  If  a  prism  of  any  form  is  subject  to  tension  T  uniform 
over  its  plane  ends,  and  no  lateral  traction,  we  have 

S  =  -  afSaQ  -  Xm  +  2n<p0. 

From  this  equation,  taking  the  first  scalar  invariant  of 
both  sides, 

T  =  3mA  +  2muh 
so  that 

rrn=  T/(3\+2fi). 

Substituting,  we  have 


2/i       v         2ju(3X  +2ju) 

We  write  now  E  =  /x(3X  +  2/x)/(X  +  /x)>  the  quotient  of  a 
simple  longitudinal  tension  by  the  stretch  produced,  and 
called  Young's  modulus.     Also  we  set 

s  =  X/(2X  +  2/x),  Poisson's  ratio, 


278  VECTOR   CALCULUS 

the  ratio  of  the  lateral  contraction  to  the  longitudinal 
stretch. 

It  is  clear  that  if  any  two  of  the  three  moduli  are  known, 
the  other  may  be  found.     We  have 

X  =  E/[(l  +  *)(1  -  2*),        M  -  \Ej(X  +  *), 
k  -  IE/(1  -  2s). 

In  terms  of  E  and  s  we  have 

t»i(S)'« 


po 


-m* 


E 


(4).  If  |  <  s,  k  <  0,  and  the  material  would  expand  under 
pressure.     If  s  <  —  1,  W  would  not  be  positive. 

(5).  If  Cauchy's  relations  hold,  s  =  \  and  X  =  /x.  For 
numerical  values  of  the  moduli  see  texts  such  as  Love, 
Elasticity. 

31.  Bodies  that  are  not  isotropic  are  called  aelotropic. 
For  discussion  of  the  cases  and  definitions  of  the  moduli, 
see  texts  on  elasticity. 

32.  There  is  still  the  problem  of  finding  a  from  cp0  after 
the  latter  has  been  found  from  S.  This  problem  we  can 
solve  as  follows: 

<t  =  <tq-\-  fp^da  =  (To  —  J£<rS  Vdp,  where  V  acts  on  a 
=  o-o  +  fgW&P  ~  hVdpVVv] 
=  *o  +  fgWdp  ~  WiPi  ~  p)VVd<r 

-d-V(Pl-  p)VV<r] 
=  <to-  Wifii  ~  Po)VVao+  fPPModp 

-iVQ>i-p)VvM 

=  <ro-  \Vifii  -  Po)VV<ro  +  fPSl[<Podp 

-  V(Pl-  p)W<Po'dp]. 

We  are  thus  able  to  express  a  at  any  point  pi  in  terms  of  the 


DEFORMABLE    BODIES  279 

values  at  p0  of  cr,  VVc,  and  the  values  along  the  path  of 
integration  of  <p0  and  FV^oO- 

EXAMPLES 

(1).  Let  us  consider  a  cylinder  or  prism  which  is  vertical 
with  horizontal  ends,  the  upper  being  cemented  to  a  hori- 
zontal plane.     Then  we  have  the  value  of 

%  =  —  gcySypSyQ,       y  vertical  unit, 

where  the  origin  is  at  the  center  of  the  lower  base.  The 
conditions  of  equilibrium  are 

S V  +  c£  =0,        or        c{  *  -  gey,        J  =  -  gy. 

That  is,  the  condition  is  realizable  by  a  cylinder  hanging 
under  its  own  weight.  The  tension  at  the  top  surface  is 
gel  where  I  is  the  length. 

Solving  for  the  strain,  we  have 

Let  a  =  gcs/E,  b  =  gc(l  +  s)IE,  and  note  that 

FWoO  =  -  aVy()  -  bVyySyQ  =  -  aVy(). 

The  integral  is  thus 

°"o  —  hV(fti  —  p0)e0 

+  fp'oiaSyp-dp  +  bySypSydp  +  aV(Pl  -  p)Vydp] 
=  (To—  \V(p\  —  po)e0 

+  Jl?[aSyp-8p  +  bySypSydp 

+«  VpiVydp  —  adpSyp  +  aySpdp] 
=  a0  —  W{pi  —  p0)e0  +  HbyS2yp 

+  aVPlVyp  +  haypX, 

the  differential  being  exact.  This  gives  us  as  the  value  of 
a  at  pi, 


280  VECTOR  CALCULUS 

*l  «  f  +  V(pi  -  p0)(ieo  +  aVypo)  +  iaFprypi 

-f  \byS2yp>       f  >  «  constants. 

Substituting  a  and  6,  and  constructing 

<Po  =  -  J[S()V-«r+  V&r()], 

we  easily  verify.  If  the  cylinder  does  not  rotate,  we  may 
omit  the  second  term  and  if  the  upper  base  does  not  move 
laterally,  then  the  vector  f  reduces  to  —  ^gcP/E-y,  and 
we  have 

'  =  -  hgcP/E-y  +  gcs/2E-Vpyp  +  gc(l  +  s)/2E-yS2py. 

A  plane  cross-section  of  the  cylinder  is  distorted  into  a 
paraboloid  of  revolution  about  the  axis  and  the  sections 
shrink  laterally  by  distances  proportional  to  their  distances 
from  the  free  end. 

(2).  If  a  cylinder  of  length  21  is  immersed  in  a  fluid  of 
density  c',  its  own  density  being  c,  the  upper  end  fixed,  p 
the  pressure  of  the  fluid  at  the  center  of  gravity,  then  we 
have  the  stress  given  by 

H  =  -  (p  +  gc'Syp)  -  g(c  -  c')(l  -  Syp)ySy, 

whence  calculating  <p0,  we  have 

<p0  =  1/E-l-  (p  +  gc'Syp)(-  1  +  2*)  -  gs(c  -  c') 

X  (1  -  Syp)]  -  ySy[g(c  -  c')(l  -  Syp)l  +  s)]/E. 
And 

a  =  f  +  Vdp  +  p[(-  1  +  2*)p  -  ^/(c  -  cO. 
-  Spyg[ce  -  s(c  +  c')]/E 

+  7lh(c  ~  c')(l  +  s)(l  -  Syp)2     ' 

+  hgp2W-s(c+c')]/E. 

(3).  What  does  the  preceding  reduce  to  if  c  =  a'?  Solve 
also  directly. 


DEFORMABLE    BODIES  281 

(4).  If  a  circular  bar  has  its  axis  parallel  to  y,  and  the 
only  stress  is  a  traction  at  each  end,  equivalent  to  couples 
of  moment  \ira*pt,  about  the  axis  of  y,  a  being  the  radius, 
that  is,  a  round  bar  held  twisted  by  opposing  couples,  we 
have 

S  =  -  lidfySnO  +  VpySyQ), 
<Po=  -  HiySpyO  +  VpySyQ], 
a  =  tVpySyp. 

Any  section  is  turned  in  its  own  plane  through  the  angle 
—  tSyp.     t  is  the  angular  twist  per  centimeter. 

(5).  The  next  example  is  of  considerable  importance,  as 
it  is  that  of  a  bar  bent  by  couples.     The  equations  are 

g  =  -  E/R-Sap-ySyQ, 

Po  -  -  (1  +  s)/R-Sap-ySyQ  ~  s/R-Sap-Q, 
a  =  iR-i-al&yp  +  sS2ap  -  sS2yap] 

+  sR~1yaS(3pSap  —  R~1ySapSyp. 

If  the  body  is  a  cylinder  or  prism  of  any  shape  with  the 
axis  y  horizontal,  there  is  no  body  force  nor  traction  on  the 
perimeter.     The  resultant  traction  across  any  section  is 

ff-  EjR-SapdA, 

which  will  equal  zero  if  the  origin  is  on  the  line  of  centroids 
of  the  sections  in  the  normal  state,  that  is,  the  neutral  axis. 
Thus  the  bar  is  stressed  only  by  the  tractions. at  its  terminal 
sections,  the  traction  across  any  section  being  equivalent 
to  a  couple. 

The  couple  becomes  one  with  axis  (3  =  ya  and  value 
EI/R,  where  7  is  the  moment  of  inertia  about  an  axis 
through  the  centroid  parallel  to  (3.  The  line  of  centroids 
is  displaced  according  to  the  law 

-  Saa  =  iR-'S'-yp, 

19 


2N2  VECTOR  CALCULUS 

so  that  it  is  approximately  the  arc  of  a  circle  of  radius  R. 
The  strain-energy  function  is  \ER~2-S2ap,  and  the  potential 
energy  per  unit  length  %EI/R2. 

For  further  discussion  see  Love,  p.  127  et  seq. 

(6).  When  E  =  -  E-Syp-OQ,  where  dy  =  0,  and  6  =  0', 
and  a  may  not  be  a  unit  vector,  show  that 

<Po  =  ~  (1  +  8)Syp-6Q  +  sSyp-mi(0), 
a  =  (1  +  8)tiSp6p  -  OpSpy]  +  mi«[-  \yp2  +  pSpy]. 

See  Love,  pp.  129-130. 

33.  We  recur  now  to  the  equation  of  equilibrium 

EV  +  cf  -  0. 

In  this  we  substitute  the  value  of 

H  =  Xmi  +  2/^o  =  -  XSVo-  -  (o-/S()  V  +  V&r()), 

whence 

XV*SVcr  +  m W  +  n\/SS7<r  -  cf  =  0, 
or 

(X  +  M)  VSVo-  +  MV2c  -  c?  =  0, 

or  equally  since 

VV  =  VSVa  +  VVVa, 
(X  +  2M)  V»SVcr  +  fiVVVcr  -  c£  =  0. 

This  is  the  equation  of  equilibrium  when  the  displacement 
and  the  force  £  are  given.  In  the  case  of  small  motion  we 
insert  on  the  right  side  instead  of  0,  —  ca".  The  traction 
across  a  plane  of  normal  v  is 

—  (X  +  iJ,)vSVcr  —  pV\Jvv, 

where  v  is  constant.  Operating  on  the  equilibrium  equa- 
tion by  *SV(),  we  see  that 

(X+2/z)V2SV<r-oSv£=  0. 


DEFORMABLE    BODIES  283 

If  then  there  are  no  body  forces  £  or  if  the  forces  £  are 
derivable  from  a  force-function  P  and  V2P  =  0  throughout 
the  body,  we  see  that 

SVa 

is  a  harmonic  function.     Since  rai(E)  =  Skmi,  we  see  that 
mi(H)  is  also  harmonic. 
Again  we  have 

(X  +  m)V#Vo-  =  -  MVV, 

whence  we  can  construct  the  operators 

(X  +  /xj V£v()£V<r  -  -  mV2V&j  -  -  Mvvsv(). 
and  adding  the  two, 

2(X  +  M)VSvSV(r()  -  -  mV2(^V()  +  V&r()) 
Now  we  have 

g  =  -  \SVct  -  m(^V()  +  V-ScrO), 
and  since  S\7<r  is  harmonic 

V2H  =  -  /xV2(^V()  +  V&r())  =  2(X  +  /*)  ViS ViSVcrO 
2(X  +  M) 


3& 
or 


V#V£V<7()  =  (1  +  s^VSvSVtrQ. 


V2H  =  ^-  ViSvifiO. 

This  relation  is  due  to  Beltrami,  R.  A.  L.  R.,  (5)  1  (1892). 

EXAMPLE 
Maxwell's  stress  system  cannot  occur  in  a  solid  body 
which  is  isotropic,  free  from  the  action  of  body  forces,  and 
slightly  strained  from  a  state  of  no  stress,  since  we  have 
-Wil(E)  =  1/8tt-(vP)2, 


284  VECTOR  CALCULUS 

which  is  not  harmonic.  (Minchin  Statics,  3d  ed.  (1886), 
vol.  12,  ch.  18.) 

34.  We  consider  now  the  problem  of  vibrations  of  a  solid 
under  no  body  forces,  the  body  being  either  isotropic  or 
aeolotropic. 

The  equation  of  vibrations  is 

c<r"  =  6(  V,  V,  <r),    where    S  =  6[(),  V,  <r]  as  before,  and 

a  is  a  function  of  both  t  and  p.  If  the  vector  co  represents 
the  direction  and  the  magnitude  of  the  wave-front,  the 
equation  of  a  plane-wave  will  be 

u  =  t  —  Sp/co, 

since  this  represents  a  variable  plane  moving  along  its 
own  normal  with  velocity  w.  By  definition  of  a  wave-front 
the  displacement  from  the  mean  position  is  at  any  instant 
the  same  at  every  point.  That  is,  a  is  a  function  of  u 
and  t,  hence 

Vo"  =  —  VSp/ooda/du  =  uT^a/du, 

and  any  homogeneous  function  of  V  as/(V)  gives 

/V-<r  =  f{oTl)dn(rldun, 

where  n  is  the  degree  of  /. 

The  equation  above  for  wave-motion  then  is 

cv"  =  e[oj-\  or1,  d2a/du2]. 

If  the  wave  is  permanent,  a  involves  t  only  through  u  and 
if  the  vibration  is  harmonic  of  frequency  p, 

<r"  =  du2a/d2  =  -  fa. 
Therefore 

e[Uu,  Uw,  a]  =  ctrT*u. 

Hence  for  a  plane  wave  propagated  in  the  direction  Uoj 


DEFORMABLE    BODIES  285 

the  vibration  is  parallel  to  one  of  the  invariant  lines  of  the 
function 

e[U<a,  Uco,  ()]. 

The  velocity  is  the  square  root  of  the  quotient  of  the  latent 
root  corresponding,  by  the  density.  There  may  be  three 
plane-polarized  waves  propagated  in  the  same  direction 
with  different  velocities.  The  wave- velocity  surface  is 
determined  by  the  equation 

S[e(w-\  co"1,  a)  -  ca][e(u-\  co"1,  (3  -  cjSHeC&T1,  co"1,  y]  =  0, 

that  is,  by  the  cubic  of  Q[Uu,  Uu,  ()]. 

If  there  is  an  energy  function,  Q[Uu,  Uu,  ()]  is  self- 
conjugate  as  may  easily  be  seen.  In  such  case  the  invariant 
lines  are  perpendicular,  that  is,  the  three  directions  of 
vibration,  0U  62,  03,  for  any  direction  of  propagation  are 
mutually  trirectangular.  Since  W  is  essentially  positive, 
the  roots  are  positive,  and  there  are  thus  three  real  velocities 
in  any  direction. 

If  g  is  a  repeated  root,  there  is  an  invariant  plane  of 
indeterminate  lines  and  the  condition  for  such  is 

V[e(«T\  to"1,  a)  -  ca][e(^-\  co"1,  0)  -  cfi]  =  0, 

a  and  /3  arbitrary.  There  is  a  finite  number  of  solutions  to 
this  vector  equation,  giving  co,  and  these  give  Hamilton's 
internal  conical  refraction.  The  vectors  terminate  at 
double  points  of  the  wave-velocity  surface. 

The  index-surface  of  MacCullagh,  that  is,  Hamilton's 
wave-slowness  surface,  is  given  by 

5[0(p,  p,  a)  -  ca][G(p,  p,  (3)  -  cj8][0(p,  p,  7)  ~  ey]  =  0, 

a,  jS,  7  arbitrary,  which  is  the  inverse  of  the  wave-velocity 
surface,  p  is  the  current  vector  of  the  surface,  just  as  co 
for  the  other  surface,  the  equation  being  formed  by  setting 


286  VECTOR  CALCULUS 

p  =  —  a>-1.  The  wave-surface,  or  surface  of  ray- velocity, 
is  the  envelope  of  Sp/o)  =  1,  or  Spp  =  —  1,  where 
/x  =  —  w_1.  The  condition  is  that  given  by  the  equations 
of  the  two  other  surfaces.  It  is  the  reciprocal  of  the  index 
surface  with  respect  to  the  unit  sphere  p2  =  —  1,  or  the 
envelope  of  the  plane  wave-fronts  in  unit  time  after  passing 
the  origin,  or  the  wave  of  the  vibration  propagated  from  the 
origin  in  unit  time.  The  vectors  p  that  satisfy  its  equation 
are  in  magnitude  and  direction  the  ray- velocities.  When 
there  is  an  energy  function,  this  ray-velocity  is  found 
easily,  as  follows: 

The  wave-surface  is  the  result  of  eliminating  between 

0(/x,  p,  a)  =  ca, 

Q(dp,  p,  a)  +  0(ju,  dp,  a)  +  0(ju,  /x,  da)  =  cdcr, 
Sup  =  -  1-Spdfi=  0. 

From  the  second  equation 

2SdfxG(<T,  a,  n)  +  SdaOiii,  fi,  a)  =  cSadX, 

or  by  the  equations 

Sdp.e(<r,  a,  /x)  =  0. 

Hence  as  dfi  is  perpendicular  to  p,  we  have 

G(<r,  <r,  p)  =  xp. 

Operate  by  Sp  and  substitute  the  value  of  x, 

Q(U<t,  Ua,  p)  =  cp. 

This  equation  with  6(p,p,  a)  =  ca  gives  all  the  relations 
between  the  three  vectors.     See  Joly,  p.  247  et  seq. 


CHAPTER  XI 
HYDRODYNAMICS 

1.  Liquids  and  gases  may  be  considered  under  the  com- 
mon name  of  fluids.  By  definition,  a  perfect  fluid  as  dis- 
tinguished from  a  viscous  fluid  has  the  property  that  its 
state  of  stress  in  motion  or  when  stationary  can  be  con- 
sidered to  be  an  operator  which  has  three  equal  roots  and 
all  lines  invariant,  thus 

E  =  -p(), 

where  p  is  positive,  that  is,  a  pressure,  or  S  =  —p.  If  the 
density  is  c,  we  have,  when  there  are  external  forces  and 
motion,  the  fundamental  equation  of  hydrodynamics 

<r"  =  J  -  c~lVp. 

In  the  case  of  viscous  fluids  we  have  to  return  to  the 
general  equation 

c  (*"  -  {)  «  -  Vp  -  (X  +  m)  VSV  o-  -mW. 

2.  When  there  is  equilibrium 

Vp  =  c£. 

If  the  external  forces  may  be  derived  from  a  force  function, 
P,  we  have  Vp  =  cVP,  hence  —  SdpVp  =  —  ScdpVP, 
or  dp  =  cdP  for  all  directions.  That  is,  any  infinitesimal 
variation  of  the  pressure  is  equal  to  the  density  into  the 
infinitesimal  variation  of  the  force  function.  In  order  that 
there  may  be  equilibrium  under  the  forces  that  reduce  to 
£,  we  must  have  £  subject  to  a  condition,  for  from  Vp  =  c£, 
we  have  V2p  =  Vc£  +  cV£,  whence  ££V£  =  0,  and 
VV%  =  F£Vlogc. 

287 


288  VECTOR    CALCULUS 

If  £  =  VP,  the  condition  is,  of  course,  satisfied,  and 
from  the  last  equation  we  see  that  £  is  parallel  to  Vc,  that 
is  to  say,  £  is  normal  to  the  isopycnic  surface  at  the  point, 
or  the  levels  of  the  force  function  are  the  isopycnic  surfaces. 
The  equation  Vp  =  c£  states  that  £  is  also  a  normal  of  the 
isobaric  surfaces.  In  other  words,  in  equilibrium  the  iso- 
baric  surfaces,  the  isopycnic  surfaces,  and  the  isosteric  sur- 
faces are  geometrically  the  same.  However,  it  is  to  be 
noted  that  if  a  set  of  levels  be  drawn  for  any  one  of  the 
three  so  that  the  values  of  the  function  represented  differ 
for  the  levels  by  a  unit,  that  is,  if  unit  sheets  are  constructed, 
then  the  levels  in  the  one  case  may  not  agree  with  the  levels 
in  the  other  two  cases  in  distribution. 

The  fundamental  equation  above  may  be  read  in  words: 
the  pressure  gradient  is  the  force  per  unit  volume.  Specific 
volume  times  pressure  gradient  is  the  force  per  unit  mass. 

We  can  also  translate  the  differential  statement  into 
words  thus:  the  mean  specific  volume  in  an  isobaric  unit 
sheet  is  the  number  of  equipotential  unit  sheets  that  are  in- 
cluded in  the  isobaric  unit  sheet.  The  average  density  in  an 
equipotential  unit  sheet  is  the  number  of  isobaric  unit  sheets 
enclosed. 

Since  dp  and  dP  are  exact  differentials,  we  have : 

Under  statical  conditions  the  line  integral  of  the  force  of 
pressure  per  unit  mass  as  well  as  the  line  integral  of  the  force 
from  the  force  function  per  unit  volume  are  independent  of  the 
path  of  integration  and  thus  depend  only  on  the  end  points. 

3.  There  is  for  every  fluid  a  characteristic  equation  which 
states  a  relation  between  the  pressure,  the  density,  and  a 
third  variable  which  in  the  case  of  a  gas  may  be  the  tempera- 
ture, or  in  the  case  of  a  liquid  like  the  sea,  the  salinity. 
Thus  the  law  of  Gay-Lussac-Mariotte  for  a  gas  is 

p  =  const  -c  (1+  t^t  T)  f°r  constant  volume. 


HYDRODYNAMICS  289 

The  characteristic  equation  usually  appears  in  the  form 
pa  =  RT,  where  in  this  case  a  is  the  specific  volume,  the 
equation  reading 

dP  =  adp. 
From  this  we  have 

dP  =  RTdp/p. 

If  T  is  connected  with  p  by  any  law  such  as  that  given 
above,  we  can  substitute  its  value  and  integrate  at  once. 
Or  if  T  is  connected  with  the  force  function  P  by  an  equa- 
tion, we  can  integrate  at  once. 

Example. 
In  the  case   of  gravity   and   the   atmosphere,   suppose 
that  the  temperature  decreases  uniformly  with   the   equi- 
potentials.     Since   we  must  in  this   case  take  P  so  that 
VP  will  be  negative,  we  have 

dP  =  -  RTdp/p,         T  =  T0  -  bP, 
whence 

dP  =  -dT/b,        dT/T  =  Rbdp/p,         T  =    T0(p/p0)bR. 

Or  again 

dP/(T0  -bP)=  -  R  dp/p,         1  -  bP/To  =  (p/po)R. 

We  thus  have  the  full  solution  of  the  problem,  the  initial 
conditions  being  for  mean  sea-level,  and  in  terms  of  a  or 
c  as  follows : 

T=  T0(p/p0)bR,         a=  a0(p/po)bR-\ 

p  =  b-iT0[i-  (p/p0n 

T=  T0(l  -  bTo-'P),       c=  c0(l  -  To-'bP)  »"1»"1-1, 
p  =  Pod-  To-'bP)*-1*-1. 

Absolute  zero  would  then  be  reached  at  a  height  where  the 


200  VECTOR    CALCULUS 

gravity  potential  would  be 

P  =  To/b, 

and  substituting  we  find  c  =  0,  p  =  0.  If  b  is  negative, 
the  fictive  limit  of  the  atmosphere  is  below  sea-level.  For 
values  of  bR  from  oo  to  1,  for  the  latter  value  b  =  0.00348 
(that  is,  a  temperature  drop  of  3.48°  C.  per  100  dynamic 
meters  of  height),  we  have  unstable  equilibrium,  since 
from  the  equations  above  for  c  we  have  increasing  density 
upwards.  The  case  bR  =  1  is  extreme;  however,  it  is 
mathematically  interesting  from  the  simplicity  that  re- 
sults. Pressure  and  temperature  would  decrease  uniformly 
and  we  should  have  a  homogeneous  atmosphere.  This 
condition  is  unstable  and  the  slightest  displacement  would 
continue  indefinitely.  Values  of  bR  less  than  1  lead  still 
to  unstable  equilibrium,  the  state  of  indifferent  equilibrium 
occurring  when  the  adiabatic  cooling  of  an  upward  moving 
mass  of  air  brings  its  temperature  to  that  of  the  new  levels. 
For  dry  air  this  occurs  for  bR  =  0.2884  =  (1.4053 
-  1)  /1.4053,  or  a  fall  of  1.0048°  C.  per  dynamic  hectom- 
eter. 

See  Bjerknes,  Dynamic  Meteorology  and  Hydrography. 

4.  The  equation  when  there  is  not  equilibrium  gives  us 

aVp  —  £  *  —  a". 

Let  £  =  VP,  and  operate  by  V*V  (),  then 

WaVp  =  -  VV<r". 

If  we  multiply  by  SUv  and  integrate  over  any  surface  nor- 
mal to  Up,  we  have 

SfSUvWaVp  =  -  ffSUvW"  =  -  fSdpa". 

The  right-hand  side  is  the  circulation  of  the  acceleration 
or  force  per  unit  mass  around  any  loop,  the  left-hand  side 


HYDRODYNAMICS  291 

is  the  surface  integral  of  WaVp  over  the  area  enclosed. 
If  then  we  suppose  that  in  a  drawing  we  represent  the  iso- 
bars as  lines,  and  the  isosterics  also  as  lines  that  cut  these, 
drawing  a  line  for  the  level  that  bounds  a  unit  sheet  in  each 
case  (and  noticing  that  in  equilibrium  the  lines  do  not  in- 
tersect), we  shall  have  a  set  of  curvilinear  parallelograms 
representing  tubes.  The  circulation  of  the  force  per  unit 
mass  around  any  boundary  will  then  be  the  number  of 
parallelograms  enclosed.  It  is  to  be  noticed  that  the  areas 
must  be  counted  positively  and  negatively,  that  is,  the 
number  of  tubes  must  be  taken  positive  or  negative,  ac- 
cording to  whether  Vfl,  Vp,  the  two  gradients,  make  a 
positive  or  a  negative  angle  with  each  other  in  the  order  as 
written.  This  circulation  of  the  force  per  unit  mass  may  be 
taken  as  a  measure  of  the  departure  from  equilibrium. 
In  the  same  way  we  find  that  if  we  draw  the  equipotentials 
and  the  isopycnics,  we  shall  have  the  number  (algebraically 
considered)  of  unit  tubes  in  any  area  equal  to  the  circula- 
tion of  the  force  per  unit  volume  around  the  bounding 
curve. 

If  we  choose  as  boundary,  for  example,  a  vertical  line,  an 
isobaric  curve,  a  downward  vertical,  and  an  isobaric  curve, 
the  number  of  isobaric-isosteric  tubes  enclosed  gives  the  differ- 
ence between  the  excess  up  one  vertical  of  the  cubic  meters 
per  ton  at  the  upper  isobar  over  that  at  the  lower  isobar  and 
the  corresponding  excess  for  the  other  vertical.  If  the  lines 
are  two  verticals  and  two  equipotentials,  the  number  of 
isopotential-isopycnic  tubes  is  the  difference  of  the  two 
excesses  of  pressure  at  the  lower  levels  over  pressure  at  the 
upper  levels.  These  are  the  circulations  around  the  bound- 
aries of  the  forces  per  unit  mass  or  unit  volume  as  the  case 
may  be. 

5.  If  we  integrate  the  pressure  over  a  closed  space  inside 


292  VECTOR    CALCULUS 

the  fluid,  we  have 

ffyUvdA  =  fffVpdv  =  fffc&v. 

But  this  latter  integral  is  the  total  force  on  the  volume 
enclosed.  This  is  Archimedes'  principle,  usually  related 
to  a  body  immersed  in  water,  in  which  case  the  statement 
is  that  the  resultant  of  all  the  pressure  of  the  water  upon 
the  immersed  body  is  equal  to  the  weight  of  the  water  dis- 
placed. If  we  were  to  consider  the  resultant  moment  of 
the  normal  pressures  and  the  external  forces,  we  would 
arrive  at  an  analogous  statement.  The  field  of  force,  how- 
ever, need  not  be  that  due  to  gravity. 

EXERCISE. 

Consider  the  case  of  a  field  in  which  there  is  the  vertical 
force  due  to  gravity  and  a  horizontal  force  due  to  centrif- 
ugal force  of  rotation. 

6.  We  turn  our  attention  now  to  moving  fluids.  A 
small  space  containing  fluid  with  one  of  its  points  at  po 
may  be  followed  as  it  moves  with  the  fluid,  always  con- 
taining the  same  particles.  It  will  usually  be  deformed  in 
shape.  The  position  p  of  the  particle  initially  at  p0  will 
be  a  function  of  p0  and  of  t,  say 

p  =  0  (p0,  t). 

The  particle  initially  at  p0  +  dp  will  at  the  same  time  t 
arrive  at  the  position 

p  +  dip  =  6  (p0  +  dp,  t)  =  p  —  SdipVo-p, 

hence  dip  becomes  at  time  t 

—  SdipVo'P  =  <pdipo. 

It  follows  that  the  area  Vdipd2p  =  V(pdip0(pd2po,  and  the 


HYDRODYNAMICS  293 

volume 

—  Sdipdtpdzp  =  —  S(pdipo(pd2po<pdspo  = 

—  SdipQd2Poddp0'  ms((p) . 

If  the  fluid  has  a  constant  mass,  then  we  must  have 

cdv  =  Codvo,     or     cra3  =  c0. 

This  is  the  equation  of  continuity  in  the  Lagrangian  form. 
The  reference  of  the  motion  to  the  time  and  the  initial  con- 
figuration is  usually   called  reference  to  the  Lagrangian 
variables. 
7.  Since 

dp  =  —  SdpVp  =  —  S<pdp0Vp 

=  —  Sdpo<p'Vp  =  —  SdpoVoP, 
VoP  =  <p'Vp  =  -  VoSpV-p. 

But  the  equations  of  motion  are  already  given  in  the  form 

aVp  =  £  -  p", 

hence  in  terms  of  the  variables  po  and  t  we  have 

aVop  =  <p'(p  —  p")- 

This  equation,  the  characteristic  equation  of  the  fluid 

F(p,  c,  T)  -  0, 

and  the  equation  of  continuity,  give  us  five  scalar  equations 
expressing  six  numbers  in  terms  of  p0  and  t.  In  order  to 
make  any  problem  definite  then,  we  must  introduce  a 
further  hypothesis.  The  two  that  are  the  most  common 
are 

(1)  The  temperature  is  constant,  if  T  is  temperature, 
or  the  salinity  is  constant,  if  T  is  salinity.  In  case  both 
variables  come  in,  we  must  have  two  corresponding  hypoth- 
eses: 


21)4  VECTOR    CALCULUS 

(2)  The  fluid  is  a  gas  subject  to  adiabatic  change. 
The  relation  of  pressure  to  density  in  this  case  is  usually 
written 

p  =  kcy. 

y  is  the  ratio  of  specific  heat  under  constant  pressure  to 
that  under  constant  volume,  as  for  example,  for  compressed 
air,  7  -  1-408. 

8.  In  the  integrations  we  are  obliged  to  pay  attention 
to  two  kinds  of  conditions,  those  due  to  the  initial  values  of 
the  space  occupied  by  the  fluid  at  t  =  0,  the  pressure  p0 
and  density  c0,  or  specific  volume  a0,  at  each  point  of  the 
fluid,  and  the  initial  velocities  of  the  particles  p0'  at  p0. 
The  other  conditions  are  the  boundary  conditions  during 
the  movement.  As  for  example,  consider  a  fluid  enclosed 
in  a  tank  or  in  a  pipe  or  conduit.  The  velocity  in  the 
latter  case  must  be  tangent  to  the  walls.  If  we  have  the 
general  case  of  a  moving  boundary  for  the  fluid,  then  its 
equation  would  be 

/(P,  t)  =  o. 

If  then  p'  is  the  velocity,  we  must  have 

-  SdPVf+  (df/dt)dt  =  0,     or  -  £p'V/+  df/dt  =  0. 

If  there  is  a  free  surface,  then  the  pressure  here  must  be 
constant,  as  the  pressure  of  the  air.  In  order  to  have 
various  combinations  of  these  conditions  coexistent,  it  is 
necessary  sometimes  to  introduce  discontinuities. 

9.  If  we  were  in  a  balloon  in  perfect  equilibrium  moving 
along  with  one  and  the  same  mass  of  air,  the  barograph 
would  register  the  varying  pressures  on  this  mass,  the  ther- 
mograph the  varying  temperatures,  and  if  there  were  a 
velocitymeter,  it  would  register  the  varying  velocity  of  the 
mass.  From  these  records  one  could  determine  graphically 
or  numerically  the  rates  of  change  of  all  these  quantities  as 


HYDRODYNAMICS  295 

they  inhere  in  the  same  mass.  That  is,  we  would  have  the 
values  of 

dp/dt,        dT/dt,        dp/dt. 

These  may  be  called  the  individual  time-derivatives  of  the 
quantities.  As  the  balloon  passed  any  fixed  station  the 
readings  of  all  the  instruments  would  be  the  same  as  instru- 
ments at  the  fixed  stations.  But  the  rates  of  change  would 
differ.  The  rates  of  change  of  these  quantities  at  the  same 
station  would  be  for  a  fixed  p  and  a  variable  t,  and  could 
be  called  the  local  time-derivatives,  or  partial  derivatives. 
They  can  be  calculated  from  the  registered  readings.  The 
relation  between  the  two  is  given  by  the  equation 

d/dt  =  d/dt  -  Sp'V. 

Thus  we  have  between  the  individual  and  the  local  values 
the  relations 

The  last  equation  gives  us  the  individual  acceleration 
in  terms  of  the  local  acceleration  and  the  velocity.  From 
the  fundamental  equation  we  have 

ovp  =  f  -  dp' let  +  w  \p'  =  i  -  dp'ldt  ~  *<j>% 

where  the  function 

0=-S()V-p',         0'=-VV(),         0o  = 

K-sovy-wo), 

2e  =  FVp'. 
This  statement  of  the  motion  in  terms  of  the  coordinates  of 


296  VECTOR    CALCULUS 

any  point  and  the  time  is  the  statement  in  terms  of  Eulers 
variables. 

Since    near  po,    p  =  po  +  po'dt,    we    have    the    former 
function  <p  at  this  point  in  the  form 

<p=  -  S()Vo-p  =  l  +  <ft(-  S()V-p')0=  1  +  d^atpo. 

Whence 

m3(<p)  =  1  +  dtmi(6)  =  1  +  dt{-  SVp'). 

Since  the  initial  point  is  any  point,  this  equation  holds  for 
any  point  and  we  have  the  equation  of  continuity  in  the 
form 

c  -  cdtSVp'  =  c0  =  c0  +  dt-dc/dt(l  ~  dtSVp'), 

or,  dropping  terms  of  second  order, 

dc/dt  -  cSVp'  =  0. 

This  is  the*  equation  of  continuity  in  the  Euler  form. 
If  we  use  local  values, 

dc/dt-  SV(cp')  =  0. 

That  is,  the  local  rate  of  change  of  the  density  is  the  con- 
vergence of  specific  momentum.  It  is  obvious  that  if  the 
fluid  is  incompressible,  that  is,  if  the  density  is  constant, 
then  the  velocity  is  solenoidal.  If  the  specific  volume  at  a 
local  station  is  constant,  then  the  specific  momentum  is 
solenoidal.  If  the  medium  is  incompressible  and  homo- 
geneous, then  both  velocity  and  specific  momentum  are 
solenoidal  vectors.  It  is  clear  also  that  in  any  case  the 
normal  component  of  velocity  must  be  continuous  through 
any  surface,  but  specific  momentum  need  not  be.  If  any 
boundary  is  stationary,  then  both  velocity  and  specific 
momentum  are  tangential  to  it. 


HYDRODYNAMICS  297 

In  the  atmosphere,  which  is  compressible,  specific  mo- 
mentum is  solenoidal,  but  in  the  incompressible  hydro- 
sphere, both  velocity  and  specific  momentum  are  solenoidal. 
Of  course  the  specific  volume  of  the  air  changes  at  a 
station,  but  only  slowly,  so  that  the  approximate  statement 
made  is  close  enough  for  meteorological  purposes. 

If  at  any  given  instant  we  draw  at  every  point  a  vector 
in  the  direction  of  the  velocity,  these  vectors  will  determine 
the  vector  lines  of  the  velocity  which  are  called  lines  of 
flow.  These  lines  are  not  made  up  of  the  same  particles 
and  if  we  were  to  mark  a  given  set  of  particles  at  any  time, 
say  by  coloring  them  blue,  then  the  configuration  of  the 
blue  particles  would  change  from  instant  to  instant  as  they 
moved  along.  The  trajectory  of  a  blue  particle  is  a  stream 
line.  If  the  particles  that  pass  a  given  point  are  all  colored 
red,  then  we  would  have  a  red  line  as  a  line  of  flow,  only  when 
the  condition  of  the  motion  is  that  called  stationary.  In 
this  case  the  line  through  the  red  particles  would  be  the 
streamline  through  the  point.  If  the  motion  is  not  sta- 
tionary, then  after  a  time  the  red  particles  would  form  a 
red  filament  that  would  be  tangled  up  with  several  stream 
lines. 

10.  In  the  case  of  meteorological  observations  the  di- 
rection of  the  wind  is  taken  at  several  stations  simultane- 
ously and  by  the  anemometer  its  intensity  is  given.  These 
data  give  us  the  means  of  drawing  on  a  chart  suitably  pre- 
pared the  lines  of  flow  at  the  given  time  of  day  and  the 
curves  showing  the  points  of  equal-intensity  of  the  wind 
velocity.  Of  course,  the  velocity  is  usually  only  the  hori- 
zontal velocity  and  the  vertical  velocity  must  be  inferred. 

One  of  the  items  needed  in  meteorological  and  other 
studies  is  the  amount  of  material  transported.  If  the  spe- 
cific momentum  in  a  horizontal  direction  is  cpr,  and  lines 

20 


298  VECTOR    CALCULUS 

of  flow  be  drawn,  then  for  a  vertical  height  dz  and  a  width 
between  lines  of  flow  equal  to  dn,  we  will  have  the  trans- 
port equal  to  Tp'dndz.  Since,  however,  we  have  for  prac- 
tical purposes  dz  =  —  dp,  we  can  write  this  in  the  form 

transport  =  Tp'dn(—  dp). 

In  order  to  do  this  graphically  we  first  draw  the  lines  of 
flow  and  the  intensity  curves.  An  arbitrary  outer  bound- 
ary curve  is  then  divided  into  intervals  of  arc  such  that 
the  projection  of  an  interval  perpendicular  to  the  nearest 
lines  of  flow  multiplied  by  the  value  of  Tp'  is  a  constant. 
Through  these  points  a  new  set  of  lines  of  flow  is  constructed. 
The  transport  between  these  lines  is  then  known  horizon- 
tally for  a  constant  pressure  drop,  by  drawing  the  intensity 
curves  that  represent  Tp'dn,  and  if  these  are  at  unit  values 
of  the  transport,  they  will  divide  th£  lines  of  flow  into  quad- 
rilaterals such  that  the  amount  of  air  transported  horizon- 
tally decreases  or  increases  by  units,  and  thus  the  vertical 
transport  must  respectively  increase  or  decrease  by  units, 
through  a  sheet  whose  upper  and  lower  surfaces  have  pies- 
sure  difference  equal  to  dp  =  —  1.  Towards  a  center  of 
convergence  the  lines  of  flow  approach  indefinitely  close. 
dn  decreases  and  it  is  clear  that  the  vertical  transport  up- 
ward increases.  There  may  be  small  areas  of  descending 
motion,  however,  even  near  such  centers.  In  this  manner 
we  may  arrive  at  a  conception  of  the  actual  movement  of 
the  air. 

Since  the  specific  momentum  is  solenoidal,  we  can  as- 
certain its  rate  of  change  vertically  from  horizontal  data. 
For 

0  =  SVcp'  =  —  dZ/dz  +  horizontal  convergence, 
or 

dZ/dz  =  horizontal  convergence  of  specific  momentum. 


HYDRODYNAMICS  299 

Substituting  the  value  of  dz,  we  have 

dZ/  (—  dp)  =  horizontal  convergence  of  velocity, 
dZ/dp  =  dTP'lds+  Tp'b. 

where  ds  runs  along  the  lines  of  flow,  and  5  is  the  diver- 
gence per  unit  ds  of  two  lines  of  width  apart  equal  to  1. 

These  considerations  enable  us  to  arrive  at  the  complete 
kinematic  diagnosis  of  the  condition  of  the  air.  On  this  is 
based  the  prognostications. 

11.  When  the  density  c  is  a  function  of  the  pressure  p, 
and  the  forces  and  the  velocities  can  be  expressed  as  gradi- 
ents, then'we  have  a  very  simple  general  case.     Thus  let 

c  =  f(p),        i  =  V«(p,  0i        p'  =  Vv(p,  t), 

and  set 

Q  =  u  —  fa&p,  then  VQ  =  £  —  aVp, 

the  equations  of  motion  are 

dp'/dt  +  0(p')  =  VQ,  or  since  p'  =  Vv, 
V[dv/dt  +  iT2Vv-  Q]  =  0. 

Hence  the  expression  in  brackets  is  independent  of  p  and 
depends  only  on  t  and  we  have 

dv/dt+iFW-  Q  =  h(t). 

We  could,  however,  have  used  for  v  any  function  differing 
from  v  only  by  a  function  of  t,  thus  we  may  absorb  the  func- 
tion of  the  right  into  v  and  set  the  right  side  equal  to  zero. 
We  thus  have  the  equations  of  motion 

dv/dt  +  JPVfl  -  Q  =  0,         dc/dt  -  SV(cVv)  =  0, 
c  =  /(p). 

From  these  we  have  v,  c,  p  in  terms  of  p  and  /. 

12.  In  the  case  of  a  permanent  motion,  the  tubes  of  flow 
are  permanent.     If  we  can  set  £  =  Vw(p),  then  we  place 


300  VECTOR    CALCULUS 

Q  =  u  —  fadp,  and  noticing  that  p'  and  Q  do  not  depend 
on  t,  we  have 

Sp'V-p'  =  -  VQ. 

If  we  operate  by  —  Sdp  =  —  S(dsUp'),  we  have 
(kSUp'Tp'VTp'  on  the  left,  since  Sp'V-Up'  =  0.  Hence 
from  this  equation  we  have  at  once 

-  SdpGTV  -  Q)  =  0. 

Hence  along  a  tube  of  flow  of  infinitesimal  cross-section 

tiy-4-a 

This  is  called  Bernoulli's  theorem.  C  is  a  function  of  the 
two  parameters  that  determine  the  infinitesimal  line  of 
flow.     Hence  along  the  same  tube  of  flow 

J(IV  -  TW)  =  Q  ~  Qo  =  u  -  u0  -  fp*  adp. 

In  the  case  of  a  liquid  a  is  constant  and  we  can  integrate 
at  once,  giving 

}ZV-  u+ap=  C. 

From  this  we  can  find  the  velocity  when  the  pressure  is 
given  or  the  pressure  when  the  velocity  is  given.  Since 
the  pressure  must  be  positive,  it  is  evident  that  the  velocity 
square  ^  2{u  +  C),  or  else  the  liquid  will  separate.  This 
fact  is  made  use  of  in  certain  air  pumps.  In  the  case  of  no 
force  but  gravity  we  have  u  =  gz, 

iTV-g*+ap=  C. 

This  is  the  fundamental  equation  of  hydraulics.  We  can- 
not enter  upon  the  further  consideration  of  it  here. 

Vortices. 
13.  In  the  case  of  p'  =  Vv  it  is  evident  that  VVpf  =  0. 
When  this  vector,  or  the  vector  e  (§9)  does  not  vanish, 


HYDRODYNAMICS  301 

there  is  not  a  velocity  potential  and  vortices  are  said  to 
exist  in  the  fluid.  It  is  obvious  that  if  a  particle  of  the 
fluid  be  considered  to  change  its  shape  as  it  moves,  then  e 
is  the  instantaneous  velocity  of  rotation.  At  any  instant 
all  the  vortices  will  form  a  vector  field  whose  lines  have  the 
differential  equation 

VdpWp'  =  0  =  SdpV'    p  -  VSp'dp; 
that  is, 

Q'dp  =  dp',        or        0'p'  =  dp'jdt, 


from  which 


p'  .  */><%'. 


These  vector  lines  are  called  the  vortex  lines  of  the  fluid. 
Occasionally  the  vortex  lines  may  be  closed,  but  as  a  rule 
the  solutions  of  such  a  differential  equation  as  the  above 
do  not  form  closed  lines,  in  which  case  they  may  terminate 
on  the  walls  of  the  containing  vessel,  or  they  may  wind 
about  indefinitely.  The  integral  of  this  equation  will 
usually  contain  t,  and  the  vortices  then  vary  with  the  time, 
but  in  a  stationary  motion  they  will  depend  only  upon  the 
point  under  consideration. 

14.  The  equations  of  motion  may  be  expressed  in  terms 
of  the  vortex  as  follows,  since 


we  have 
and  thus 


Vp'VVp' '  =  Sp'V-p'-iVp'2,. 


Sp'V-p'  =2Vp'e  +  iVp'\ 


aVp  =  i  -  dp' Idt  +  JVp/2  +  2Vp'e. 


15.  When    now  £  =  \/u{p,  t),    and  c  =  f(p),   we   set 
P  =  fadp,  giving  VP  =  aVp,  and  thence 

VP  =  Vu  -  dp' Idt  +  JVp/2  -  2Vep'. 


302  VECTOR    CALCULUS 

Or,  if  we  set  II  =  u-\-  Jp'2  —  P,  we  have 

dp'/dt  +  2VeP'  =  VII. 

Operate  on  this  with  V-V(),  and  since  VV dp'/dt 
=  2de/dt,  and  WVep'  =  SeV  -p'  -  eSVp'  -  Sp'V-e,  de/dt 
—  Sp'V-e  =  de/dt,  SVp'  by  the  continuity  equation  is 
equal  to  c~ldc/dt  =  —  a~lda/dt,  we  have 

d(ae)/dt  =  -  S(ae)V-p'  =  6(ae). 

This  equation  is  due  to  Helmholtz. 

If  we  remember  the  Lagrangian  variables,  it  is  clear  that 
6  is  a  function  of  the  initial  vector  p0  and  of  t,  hence  the 
integral  of  this  equation  will  take  the  form 

ae  =  efm'a,eQ  =  e' ~  s^^'dta0e0  =  ^(t)a0e0. 

But  the  operator  is  proved  below  to  be  equal  to  <p  itself, 
so  that  when  £  =  Vu, 

ae  =  a0Se0Vo-p  =  +  ao<p€0, 

or  finally  we  have,  if  we  follow  the  stream  line  of  a  particle, 
which  was  implied  in  the  integration  above,  Cauchy's  form 
of  the  integral 

(a/a0)e  =  —  *Se0Vo'jP, 

where  p  is  a  function  of  p0  and  t.  It  is  evident  now  if 
for  any  particle  e  is  ever  zero,  that  is,  e0  =  0,  that  always 
e  =  0.  This  is  equivalent  to  Lagrange's  theorem  that  if 
for  any  group  of  particles  of  the  fluid  we  have  a  velocity 
potential,  then  that  group  will  always  possess  a  velocity 
potential.  (It  is  to  be  noted  that  velocity  potential  and 
vortex  are  phenomena  that  belong  to  the  particles  and  the 
stream  lines,  and  not  to  the  points  of  space  and  the  lines 
of  flow.)  It  must  be  remembered  too,  that  this  result 
was  on  the  supposition  that  the  density  was  a  function  of 


HYDRODYNAMICS  303 

the  pressure  alone,  and  that  the  external  forces  £  were 
conservative. 

16.  We  may  deduce  the  equation  above  as  follows,  which 
reproduces  in  vector  form  the  essential  features  of  Cauchy's 
demonstration.     (Appell,  Traite  de  Mec.  Ill,  p.  332.) 

Let  dp/dt  =  a,  and  Q  =  u  —  fadp,  then,  remembering 
that  Q  is  a  function  of  p  and  t,  and  p  is  a  function  of  p0 
and  /, 

da/dt  =  VQ(p,  t). 

Also  VoQ(po,  t)  =  -  VoSpVQ  =  —  VoSpda/dt,  where  Vo 
operates  on  p  only;  or  we  can  write 

VoQ  =  <p' da/dt. 

Hence,  operating  with  FVo( ),  we  have  V\7o(pf da/dt  =  0  = 
d/dt(VVo<p'a).  Thus  the  parenthesis  equals  its  initial 
value,  that  is,  since  the  initial  value  of  cp'a  is  a0,  and  since 
Vo  =  <p'V, 

VVo<p'<r  =  2e0  =  V<p'Vv'<r  =  mz{<p)<p~lVS7a  =  2m3(p~1e. 

Thus  we  have  at  once  m3e  =  (pe0.  This  is  the  same  as  the 
other  form,  since  ra3  =  a/aQ.  This  equation  shows  the 
kinematical  character  of  e,  and  that  no  forces  can  set  up  e 
or  destroy  it. 

17.  The  circulation  at  a  given  instant  of  the  velocity 
along  any  loop  is 

I  =  -  fSdpp'. 

The  time  derivative  of  this  is  dl/dt  =  tf^SdpS/Sp'p' 
-  Sdpp")  =  £(-  SdpW  tip'2  -  Q]  ).  But  this  is  an  inte- 
gral of  an  exact  differential  and  vanishes.  Hence  if  the 
forces  are  conservative  and  the  density  depends  on  the 
pressure,  the  circulation  around  any  path  does  not  change 
as  the  particles  of  the  path  describe  their  stream  lines.     The 


304  VECTOR    CALCULUS 

circulation  is  an  integral  invariant.  This  theorem  is  due  to 
Lagrange.     If  we  express  the  circulation  in  the  form 

I  - '  -  ffSdvVp'  =  -  2ffSdpe, 

we  see  that  the  circulation  is  twice  the  flux  of  the  vortex 
through  the  loop.  Hence  as  the  circulation  is  constant, 
the  flux  of  the  vortex  through  the  surface  does  not  vary 
in  time,  if  the  surface  is  bounded  by  the  stream  loop.  The 
flux  of  the  vortex  through  any  loop  at  a  given  instant  is 
the  vortex  strength  of  the  surface  enclosed  by  the  loop. 
If  a  closed  surface  is  drawn  in  the  fluid,  the  flux  through  it 
is  zero,  since  the  vortex  is  a  solenoidal  vector. 

18.  If  we  take  as  our  closed  surface  a  space  bounded  by  a 
vortex  tube  and  two  sections  of  the  tube,  since  the  surface 
integral  over  the  walls  of  the  tube  is  zero,  it  follows  that 
the  flux  of  the  vortex  through  one  section  inwards  equals 
that  over  the  other  section  outwards.  Combining  these 
theorems,  it  is  evident  that  the  vortex  strength,  or  wr- 
ticity,  of  a  vortex  tube  is  constant.  Thus  the  collection  of 
particles  that  make  up  the  vortex  tube  is  invariant  in  time. 
In  a  perfect  fluid  a  vortex  tube  is  indestructible,  and  one 
could  not  be  generated. 

19.  It  is  evident  from  what  precedes  that  a  vortex  tube 
cannot  terminate  in  the  fluid  but  must  end  either  at  a  wall 
or  a  surface  of  discontinuity,  or  be  a  closed  tube  with  or 
without  knots,  or  it  may  wind  around  infinitely  in  the  fluid. 

If  a  vortex  tube  is  taken  with  infinitesimal  cross-section, 
it  is  called  a  vortex  filament. 

20.  We  consider  next  the  problem  of  determining  the 
velocity  when  the  vortex  is  known.  That  is,  given  e,  to 
find  a  =  p'.  We  consider  first  the  case  of  an  incompressible 
fluid,  in  which  the  velocity  is  solenoidal,  that  is,  SVcr  =  0. 
This  with  the  equations  at    the  boundaries  gives   us   the 


HYDRODYNAMICS  305 

following  problem :  to  find  a  when  2e  =  FVo",  SVer  =  0, 
SUva  =  0  at  the  boundaries,  or  if  infinite  aa  =  0.  This 
problem  has  a  unique  solution,  if  the  containing  vessel  is 
simply  connected.  We  cannot  enter  extensively  into  it, 
for  it  involves  the  theory  of  potential  functions,  and  may 
be  reduced  to  integral  equations.  However,  since  SVv  =  0, 
we  may  set  a  =  VVr,  where  *$Vr  =  0,  whence 

V2r  =  2e, 

and  we  may  suppose  r  is  known,  in  the  form 

T  =  h7ffSfe/T(p-  Po)dv. 

If  we  operate  upon  this  by  FV( ),  we  find  a  formula  for  a, 

a  =  H,2ir-fffVe(p  -  p0)/T\p  -  Po)dv. 

As  we  see,  this  formula  is  capable  of  being  stated  thus: 
the  velocity  is  connected  with  its  vortex  in  the  same  way 
as  a  magnetic  field  is  connected  with  the  electric  current 
density  that  produces  it,  the  vortex  filament  taking  the 
place  of  the  cm  rent,  the  strength  of  current  being  Tej2ir, 
and  the  elements  of  length  of  the  tube  acting  like  the  ele- 
ments of  current.  This  solution  holds  throughout  the 
entire  fluid,  even  at  points  outside  the  space  that  is  actually 
in  motion  with  a  vortex. 

Since  the  equation  of  the  surface  of  the  tube  can  be 
written  in  the  form 

F(P,  t)  =  0, 

this  surface  will  move  in  time.  Its  velocity  of  displace- 
ment is  defined  like  that  of  any  discontinuity,  as 
UvFdF/dt.  On  one  side  the  velocity  is  irrotational,  on 
the  other  it  is  vortical.  On  the  irrotational  side  we  have 
the  velocity  of  the  form  a  —  V?,  and  we  must  have  on 


306  VECTOR    CALCULUS 

that  side  the  same  velocity  of  displacement  in  the  form 

UpSUpVP. 

The  energy  involved  in  a  vortex  on  account  of  the  velocity 
in  the  particles  is 

K  *  -  \cfffp'2dv 
=  "  hcfffSp'Vrdv 
=  ¥fff  [SV(p'r)  -  2Sre]dv 
=  hcffSdvp'r  -  cfffSredv 
=  —  cj  J 'fSredv     over  all  space 
=  c/2T.SffSSSSee'lT(p  -  p0)dvdv'. 

This  is  the  same  formula  as  that  of  the  energy  of  two  cur- 
rents. In  the  expression  every  filament  must  be  considered 
with  regard  to  every  other  filament  and  itself. 

Examples.  (1).  Let  there  be  first  a  straight  voitex  fila- 
ment terminating  at  the  top  and  bottom  of  the  fluid.  Let 
all  the  motion  be  parallel  to  the  horizontal  bottom.     Then 

Sya  =  0,         Vye  =  0,         de/dt  =  0. 

We    have    then 

a  =  VyVw,  2e  =  —  yV2w  =  2zy, 
say, 

w  =  —  7r    lffz  log  rdA. 

For  a  single  vortex  filament  of  cross-section  dA  and  strength 
k  =  zdA,  we  have 

iv  =  —  k/w  log  r  =  —  kjir  log  V   (#2  +  2/2) 
a=  Vy(p-  po)IT>(p-  p0).k/T, 

where  p  is  measured  parallel  to  the  bottom. 

The  velocity  is  tangent  to  the  circles  of  motion  and  in- 
versely as  the  distance  from  the  vortex  filament.  The 
motion  is  irrotational  save  at  the  filament  itself. 


HYDRODYNAMICS  307 

For  the  effect  of  vortices  upon  each  other,  and  their 
relative  motions,  see  Webster,  Dynamics,  p.  518  et  seq. 

(2).  For  the  case  of  a  vortex  ring  or  a  number  of  vortex 
rings  with  the  same  axis,  see  Appell,  Traite,  vol.  Ill,  p.  431 
et  seq. 

21.  In  the  more  general  case  in  which  the  fluid  is  com- 
pressible we  must  resort  to  the  theorem  that  any  vector 
can  be  decomposed  into  a  solenoidal  part  and  a  lamellar 
part  and  these  may  then  be  found.  The  extra  term  in  the 
electromagnetic  analogy  would  then  be  due  to  a  perma- 
nent distribution  of  magnetism  as  well  as  that  arising  fiom 
the  current. 

EXERCISES 

1.  If  Sea  =  0,  then  it  is  necessary  and  sufficient  that  a  =  M\/P, 
M  being  a  function  of  p. 

2.  Discuss  the  case  Vae  =  0.  Beltrami,  Rend.  R.  1st.  Lomb.  (2) 
22,  fasc.  2. 

3.  Discuss  Clebsch's  transformation  in  which  we  decompose  <r  thus, 
o-  =  Vm  +  lVV.  Show  that  the  vortex  lines  are  the  intersections  of 
the  surfaces  I  and  v,  and  that  the  lines  of  flow  form  with  the  vortex  lines 
an  orthogonal  system  only  when  the  surfaces  I,  u,  v  are  triply  orthog- 
onal. 

4.  Discuss  the  problem  of  sources  and  sinks. 

5.  Consider  the  problem  of  multiply-connected  surfaces,  containing 
fluids. 

22.  It  will  be  remembered  that  Helmholtz's  theorem 
was  for  the  case  in  which  the  impressed  forces  had  a  poten- 
tial and  the  density  was  a  function  of  the  pressure.  In 
this  case  we  will  have  the  equation 

da/dt  +  2Vea  =  {  -  aVp  +  JVtf2. 

Operate  by  |FV( )  and  notice  that 

de/dt  -  eSVa  -  SaV-e  =  a  -ld(ae)/dt, 

whence  we  have  the  generalized  form 

a-ld(ae)ldt  +  SeV -<r  =  iVV£  -  fFVaVp. 


308  VECTOR    CALCULUS 

If  now  at  the  instant  t0  the  particle  does  not  rotate  and  if  a 
is  a  function  of  p  alone,  then  at  this  instant  de/dt  =  JFV£, 
and  the  paiticle  will  acquire  an  instantaneous  increase  of 
its  zero  vortex  equal  to  the  vortex  of  the  impressed  force. 
That  is,  £  must  be  peimanently  equal  to  zero  if  there  is  to 
be  no  rotation  at  any  time. 

If  FV£  =  0  but  a  is  not  a  function  of  p  alone,  then  we 
have 

a-1d(ae)/dt  +  SeV -<r  =  -  §WaVp. 

The  right  side  is  a  vector  in  the  direction  of  the  intersection 
of  the  isobaric  and  the  isosteric  surfaces.  Now  if  we  take 
an  infinitesimal  length  along  the  vortex  tube,  I,  the  cross- 
section  being  A,  the  vorticity  is  ATe  =  m,  the  mass  is 
cAl  =  constant  =  M.  Then  we  have,  since  ae  =  AlejM 
=  mlUejM, 

-  SeV-<r  =  md(lUe)dtaM  «  -  ~  fUeV*  -   ^^-elf 

I  I      at 

a-1d(ae)/dt+  SeV -a  = 

dmldt-lUe/aM'+  md{We)la  Mdt  -  md(lUe)/dtaM  = 

dm/dt-lUe/aM  =  Ve-dTe/dt  = 

±|  number  of  tubes. 

Hence  the  moment  m  of  the  vortex  will  usually  change 
with  the  time  unless  the  surfaces  coincide.  Thus  a  rotat- 
ing particle  may  gain  or  lose  in  vorticity.  If  then  the 
isobaric  and  isosteric  surfaces  under  the  influence  of  heat 
conditions  intersect,  vortices  will  be  created  along  the  lines 
of  intersections  of  the  surfaces  and  these  will  persist  until 
the  surfaces  intersect  again,  save  so  far  as  viscosity 
interfeies. 

23.  Finally  we  consider  the  conditions  that  must  be 
put  upon  surfaces  of  discontinuity,  in  this  case  of  the  first 
order  in  <r,  that  is,  a  wave  of  acceleration. 


HYDRODYNAMICS  309 

Let  c  be  a  function  of    p    only.     Then 

a\/p  =  dp/dc  \7log  c,  and  the  equation  of  motion  becomes 
p"  =  J  —  dp/dc  •  V  log  c. 

Let  the  equation  of  the  surface  of  discontinuity  be  f(p0,  t) 
=  0,  the  normal  v.  Let  £,  a,  p,  and  c  be  continuous  as 
well  as  dp/dc,  but  p"  =  a'  be  discontinuous  at  the  suiface. 
Then  on  the  two  sides  of  the  surface  we  have  the  jump, 

by  p.  263, 

\p"\  =  -  dp/dc[V  log  c], 
or 

G2ix=  dp/dc -UVfSfiUVf. 

It  follows,  therefore,  that  we  must  have  V/iUVf  =  0  and 
G  =  V (dp/dc),  or  else  we  have  G  =  0  and  SnUVf  =  0. 
In  the  first  case  the  discontinuity  is  longitudinal,  in  the 
second  transversal.  This  is  Hugoniot's  theorem.  In  full 
it  is: 

In  a  compressible  but  non- viscous  fluid  there  are  possible 
only  two  waves  of  discontinuity  of  the  second  order;  a 
longitudinal  wave  propagated  with  a  velocity  equal  to 
V (dp/dc),  and  a  transversal  wave  which  is  not  propagated 
at  all. 

The  formula  for  the  velocity  in  the  first  case  is  due  to 
Laplace.  Also  we  have  for  the  longitudinal  waves  [&Vo"] 
=  —  GSfxUVf,  for  transversal  waves  equal  to  zero.  On 
the  other  hand,  for  longitudinal  waves,  [FVo\|  =  0,  for 
transversal,  =  GVUVf^. 


310  VECTOR  CALCULUS 


REFERENCES. 

1.  Mathematische  Schriften  (Ed.  Gerhart).  Berlin,  1850.  Bd.  II, 
Abt.  1,  p.  20. 

2.  On  a  new  species  of  imaginary  quantities  connected  with  a  theory 
of  quaternions.     Proc.  Royal  Irish  Academy,  2  (1843),  pp.  424-434. 

3.  Die  lineale  Ausdehnungslehre.     Leipzig,  1844. 

4.  Gow:   History  of  Greek  Mathematics,  p.  78. 

5.  Ars  Magna,  Nuremberg,  1545,  Chap.  37;  Opera  4,  Lyon,  1663,  p. 
286. 

6.  Algebra.     Bologna,  1572,  pp.  293-4. 

7.  Om  Directiones  analytiske  Betejning.  Read  1797.  Nye  Samm- 
lung  af  det  kongelige  Danske  Videnskabernes  Selskabs  Skrifter,  (2) 
5  (1799),  pp.  469-518.  Trans.  1897.  Essai  sur  la  representation  de 
la  direction,  Copenhagen. 

8.  Essai  sur  une  maniere  de  repr6senter  les  quantites  imaginaires 
dans  les  constructions  g6ometriques.     Paris,  1806. 

9.  Theoria  residuorum  biquadraticum,  commentation  secunde.     1831. 

10.  Annales  Math,  pures  et  appliqu6es.     4  (1814-4),  p.  231. 

11.  Theory  of  algebraic  couples,  etc.     Trans.  Royal  Irish  Acad.,  17 
(1837),  p.  293. 

12.  Ueber  Functionen  von  Vectorgrossen  welche  selbst  wieder  Vector- 
grossen  sind.     Math.  Annalen,  43  (1893),  pp.  197-215. 

13.  Grundlagen  der  Vektor-und  Affinor- Analysis.     Leipzig,  1914. 
\\.  Lectures  on  Quaternions.     Preface.     Dublin,  1853. 

15.  Note  on  William  R.  Hamilton's  place  in  the  history  of  abstract 
group  theory.     Bibliotheca  Mathematica,  (3)  11   (1911),  pp.  314-5. 

16.  Leipzig,  1827. 

17.  Leipzig. 

18.  Elements  of  Vector  Analysis  (1881-4),  New  Haven.  Vol.  2, 
Scientific  Papers. 


INDEX. 


Acceleration 27 

Action :    14,  28 

Activity 15,  129,  142 

Activity-density 15,  131 

Algebraic  couple 4,  65 

Algebraic  multiplication 9 

Alternating  current 71 

Ampere 30 

Anticyclone 47 

Area 142 

Areal  axis 198 

Argand 4 

Ausdehnungslehre 3,  9 

Average  velocity 57 

Axial  vector 30 

Barycentric  calculus 8 

Bigelow 50,  60 

Biquaternions 3,  126 

Biradials 94 

Bivector 29 

Bjerknes 48,  57,  59,  290 

Cailler 2 

Cardan 3 

Center  (singularity) 44 

Center  of  isogons 48 

Change  of  basis 54 

Characteristic  equation 125 

Characteristic  equation  of 

dyadic 221 

Chi  of  dyadic 235 

Christoffel's  conditions 266 

Circuital  derivative 167 

Circular  multiplication 9 

Circulation 78,  129 

Clifford 3,  90 

Combebiac 3 

Complex  numbers 63 

Congruences 51,  138 

Conjugate 66 

Conjugate  function 5 

Continuous  group 195 

Continuous  plane  media 87 

Convergence 177 

Coulomb 13 

Couple 139 


Crystals 109 

Cubic  dilatation 258 

Curl 76,82,  184 

Curl  of  field 77 

Curvature 148,  152 

Curves 148 

Cycle 30,  37 

Cyclone 47 

Derivative  dyad 242 

Developables 150 

Dickson 105 

Differential  of  p  . .  . ' 145 

Differential  of  q 155,  159 

Differential  of  vector 55 

Differentiator 248 

Directional  derivative 166 

Discharge 130 

Discontinuities 261 

Dissipation  (plane) 84 

Dissipation,  dispersion 180 

Divergence 76,  82 

Divergence  of  field 77 

Dyadic 2,  11,  218 

Dyadic  field 246 

Dyname .       2 

Dyne 29 

Electric  current 30 

Electric  density  current 30 

Electric  induction 32 

Electric  intensity 31,  139 

Energy 14 

Energy  current 30 

Energy-density 15,  131 

Energy-density  current 30 

Energy  flux 142 

Equation  of  continuity 87 

Equipollences 71 

Equipotential 15 

Erg 14 

Euler 107 

Exact  differential 190 

Exterior  multiplication 9 

Extremals 160 

Eye  of  cyclone 47 

311 


:;il> 


VECTOR  CALCULUS 


Farad 32,  73 

Faux 37,  38 

Faux-focus 44 

Feuille 30 

Feuillets 2 

Field 13 

Flow 142 

Flux 29,  130,  142 

Flux  density 29 

Focus 41 

Force 29 

Force  density 28,  141 

Force  function 18 

Franklin 90 

Free  vector 8,  25 

Frenet-vSerret  formulae 148 

Functions  of  dyadic 238 

Function  of  flow 88 

Functions  of  quaternions. ...    121 

Gas  defined 87 

Gauss 4/ 

Gauss  (magnetic  unit).  . .  .32,  130 

Gaussian  operator 108 

General  equation  of  dyadic .  .   220 

Geometric  curl 76 

Geometric  divergence 76 

Geometric  loci 133 

Geometric  vector 1 

Geometry  of  lines 2 

Gibbs 2,  11,  215 

Gilbert 32,  130,  143 

Glissant 26 

Gradient 16,  163 

Gram 15 

Grassmann 2,  3,  9 

Green's  Theorem 205 

Groups 8 

Guiot 138 

Hamilton 2,  3,  4,65,95 

Harmonics 84,  169 

Heaviside 31 

Henry  (electric  unit) 32,  73 

Hertzian  vectors 33 

Hitchcock 49 

Hodograph 27 

Hypernumber 3,  94 

Imaginary 65 

Impedance 73 

Inductance. 73 

Inductivity 32 

Integral  of  vector 56 


Integrating  factor 191 

Integration  by  parts 198 

Interior  multiplication 10 

Invariant  line 219 

Irrotational 88 

Isobaric 15,288 

Isogons 34 

Isohydric 15 

Isopycnic 15,  288 

Isosteric 15,  288 

Isothermal 15 

Joly 138,  147 

Joule 14 

Joule-second 14 

Kinematic  compatibility ....  266 

Kirchoff's  laws .' 73 

Koenig 198,  205 

Laisant 71 

Lamellae 15 

Lamellar  field 84,  181 

Laplace's  equation 214 

Latent  equation 220 

Laws  of  quaternions 103 

Leibniz 3 

Level 15 

Line  (electric  unit) 32,  130 

Lineal  multiplication 9 

Linear  associative  algebra ...  3 

Linear  vector  function 218 

Line  of  centers 46 

Line  of  convergence 47 

Line  of  divergence 47 

Line  of  fauces 46 

Line  of  foci 46 

Line  of  nodes 45 

Lines  as  levels 80 

Liquid  defined 87 

MacMahon 75 

Magnetic  current 31 

Magnetic  density  current 31 

Magnetic  induction 32 

Magnetic  intensity 32,  139 

Mass 15 

Matrix  unity 65 

Maxwell 13 

McAulay 3 

Mobius 8 

Modulus 66 

Moment 138 

Moment  of  momentum 139 


INDEX 


313 


Momentum 28 

Momentum  density 28 

Momentum  of  field 141 

Monodromic 14 

Monogenic 89 

Moving  electric  field 140 

Moving  magnetic  field 140 

Multenions 3 

Multiple 6 

Mutation 108 

Nabla  as  complex  number. . .  82 

Nabla  in  plane 80 

Nabla  in  space 162 

Neutral  point 47 

Node 37,38 

Node  of  isogons 48 

Non-degenerate  equations .  .  .  225 

Norm 66 

Notations 

One  vector 12 

Scalar 127 

Two  vectors 136 

Derivative  of  vectors 165 

Divergence,  vortex,  deriva- 
tive dyads 179 

Dyadics 248 

Ohm  (electric  unit) 73 

Orthogonal  dyadic 241 

Orthogonal  transformation  .  .     55 

Peirce,  Benjamin 3 

Peirce,  B.  O 85 

Permittance 73 

Permittivity 32 

Phase  angle 71 

Plane  fields.. 84 

Poincare 36,  46 

Polar  vector 30 

Polydromic 14 

Potential ,. .. 15,  17 

Progressive  multiplication ...      10 

Power 76 

Poynting  vector 141 

Pressure 142 

Product  of  quaternions 98 

Product    of    several    quater- 
nions     113 

Product  of  vectors 101 

Quantum 14 

Quaternions 2,  3,  6,  7,  95 


Radial 26 

Radius  vector 26 

Ratio  of  vectors 62 

Reactance 73 

Real 65 

Reflections \  108 

Refraction . . 112 

Regressive  multiplication. ...  10 

Relative  derivative 18 

Right  versor 96 

Rotations 108 

Rotatory  deviation 175 

Saint  Venant's  equations. . .  .  260 

Sandstrom 35,  49 

Saussure 2 

Scalar 13 

Scalar  invariants 220,  239 

Scalar  of  q 96 

Schouten 7 

Science  of  extension 2 

Self  transverse 234 

Servois 4 

Shear 256 

Similitude 242 

Singularities  of  vector  lines .  .  244 

Singular  lines 45 

Solenoidal  field 84,  181 

Solid  angles 117 

Solution  of  equations 123 

Solution  of  differential  equa- 
tions   195 

Solution  of  linear  equation. .  .  229 

Specific  momentum 28 

Spherical  astronomy 110 

Squirt 90 

Steinmetz 68,  71 

Stoke's  theorem 200 

Strain 253 

Strength  of  source  or  sink  ...  90 

Stress 143,269 

Study .  .  . 2 

Sum  of  quaternions 96 

Surfaces 151 

Symmetric  multiplication ...  9 

Tensor 65 

Tensor  of  q 96 

Torque 140 

Tortuosity 149 

Trajectories 150 

Transport 130,298 

Transverse  dyadic 231 

Triplex 25 


314 


VECTOR  CALCULUS 


Triquaternions 3 

Trirectangular  biradials 100 

Unit  tube 18 

Vacuity 220 

Vanishing  invariants 240 

Variable  trihedral 172 

Vector 1 

Vector  calculus 1,  25 

Vector  field 23,  26 

Vector  lines 33 

Vector  of  q 96 

Vector  potential 33,  93,  181 

Vector  surfaces 34 

Vector  tubes 34 


Velocity 27 

Velocity  potential 18 

Versor 65 

Versor  of  q 96 

Virial 129 

Volt 31,  130,  143 

Vortex 92,  187,  187 

Vorticity 247,304 

Waterspouts 50 

Watt 15 

Weber 14 

Wessel 4 

Whirl 90 

Zero  roots  of  linear  equations.  230 


foist    r       Ot—       C/       p^V      A^y 


14  DAY  USE 

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